Partial Differential Equations

2015

• Paper 1, Section II, E

  Partial Differential Equations | Part II, 2015

  (a) State the Cauchy-Kovalevskaya theorem, and explain for which values of $a \in \mathbb{R}$ it implies the existence of solutions to the Cauchy problem

  $$x u_{x}+y u_{y}+a u_{z}=u, \quad u(x, y, 0)=f(x, y),$$

  where $f$ is real analytic. Using the method of characteristics, solve this problem for these values of $a$, and comment on the behaviour of the characteristics as $a$ approaches any value where the non-characteristic condition fails.

  (b) Consider the Cauchy problem

  $$u_{y}=v_{x}, \quad v_{y}=-u_{x}$$

  with initial data $u(x, 0)=f(x)$ and $v(x, 0)=0$ which are $2 \pi$-periodic in $x$. Give an example of a sequence of smooth solutions $\left(u_{n}, v_{n}\right)$ which are also $2 \pi$-periodic in $x$ whose corresponding initial data $u_{n}(x, 0)=f_{n}(x)$ and $v_{n}(x, 0)=0$ are such that $\int_{0}^{2 \pi}\left|f_{n}(x)\right|^{2} d x \rightarrow 0$ while $\int_{0}^{2 \pi}\left|u_{n}(x, y)\right|^{2} d x \rightarrow \infty$ for non-zero $y$ as $n \rightarrow \infty$

  Comment on the significance of this in relation to the concept of well-posedness.

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• Paper 2, Section II, E

  Partial Differential Equations | Part II, 2015

  Prove that if $\phi \in C\left(\mathbb{R}^{n}\right)$ is absolutely integrable with $\int \phi(x) d x=1$, and $\phi_{\epsilon}(x)=\epsilon^{-n} \phi(x / \epsilon)$ for $\epsilon>0$, then for every Schwartz function $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$ the convolution

  $$\phi_{\epsilon} * f(x) \rightarrow f(x)$$

  uniformly in $x$ as $\epsilon \downarrow 0$.

  Show that the function $N_{\epsilon} \in C^{\infty}\left(\mathbb{R}^{3}\right)$ given by

  $$N_{\epsilon}(x)=\frac{1}{4 \pi \sqrt{|x|^{2}+\epsilon^{2}}}$$

  for $\epsilon>0$ satisfies

  $$\lim _{\epsilon \rightarrow 0} \int_{\mathbb{R}^{3}}-\Delta N_{\epsilon}(x) f(x) d x=f(0)$$

  for $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$. Hence prove that the tempered distribution determined by the function $N(x)=(4 \pi|x|)^{-1}$ is a fundamental solution of the operator $-\Delta .$

  [You may use the fact that $\int_{0}^{\infty} r^{2} /\left(1+r^{2}\right)^{5 / 2} d r=1 / 3 .$ ]

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• Paper 3, Section II, E

  Partial Differential Equations | Part II, 2015

  (a) Show that if $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$ is a Schwartz function and $u$ is a tempered distribution which solves

  $$-\Delta u+m^{2} u=f$$

  for some constant $m \neq 0$, then there exists a number $C>0$ which depends only on $m$, such that $\|u\|_{H^{s+2}} \leqslant C\|f\|_{H^{s}}$ for any $s \geqslant 0$. Explain briefly why this inequality remains valid if $f$ is only assumed to be in $H^{s}\left(\mathbb{R}^{n}\right)$.

  Show that if $\epsilon>0$ is given then $\|v\|_{H^{1}}^{2} \leqslant \epsilon\|v\|_{H^{2}}^{2}+\frac{1}{4 \epsilon}\|v\|_{H^{0}}^{2}$ for any $v \in H^{2}\left(\mathbb{R}^{n}\right)$.

  [Hint: The inequality $a \leqslant \epsilon a^{2}+\frac{1}{4 \epsilon}$ holds for any positive $\epsilon$ and $a \in \mathbb{R} .$ ]

  Prove that if $u$ is a smooth bounded function which solves

  $$-\Delta u+m^{2} u=u^{3}+\alpha \cdot \nabla u$$

  for some constant vector $\alpha \in \mathbb{R}^{n}$ and constant $m \neq 0$, then there exists a number $C^{\prime}>0$ such that $\|u\|_{H^{2}} \leqslant C^{\prime}$ and $C^{\prime}$ depends only on $m, \alpha,\|u\|_{L^{\infty}},\|u\|_{L^{2}}$.

  [You may use the fact that, for non-negative $s$, the Sobolev space of functions

  $$\left.H^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^{2}\left(\mathbb{R}^{n}\right):\|f\|_{H^{s}}^{2}=\int_{\mathbb{R}^{n}}\left(1+\|\xi\|^{2}\right)^{s}|\hat{f}(\xi)|^{2} d \xi<\infty\right\} .\right]$$

  (b) Let $u(x, t)$ be a smooth real-valued function, which is $2 \pi$-periodic in $x$ and satisfies the equation

  $$u_{t}=u^{2} u_{x x}+u^{3}$$

  Give a complete proof that if $u(x, 0)>0$ for all $x$ then $u(x, t)>0$ for all $x$ and $t>0$.

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• Paper 4, Section II, E

  Partial Differential Equations | Part II, 2015

  (a) Show that the Cauchy problem for $u(x, t)$ satisfying

  $$u_{t}+u=u_{x x}$$

  with initial data $u(x, 0)=u_{0}(x)$, which is a smooth $2 \pi$-periodic function of $x$, defines a strongly continuous one parameter semi-group of contractions on the Sobolev space $H_{\text {per }}^{s}$ for any $s \in\{0,1,2, \ldots\}$.

  (b) Solve the Cauchy problem for the equation

  $$u_{t t}+u_{t}+\frac{1}{4} u=u_{x x}$$

  with $u(x, 0)=u_{0}(x), u_{t}(x, 0)=u_{1}(x)$, where $u_{0}, u_{1}$ are smooth $2 \pi$-periodic functions of $x$, and show that the solution is smooth. Prove from first principles that the solution satisfies the property of finite propagation speed.

  [In this question all functions are real-valued, and

  $$H_{\text {per }}^{s}=\left\{u=\sum_{m \in \mathbb{Z}} \hat{u}(m) e^{i m x} \in L^{2}:\|u\|_{H^{s}}^{2}=\sum_{m \in \mathbb{Z}}\left(1+m^{2}\right)^{s}|\hat{u}(m)|^{2}<\infty\right\}$$

  are the Sobolev spaces of functions which are $2 \pi$-periodic in $x$, for $s=0,1,2, \ldots]$

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2014

• Paper 1, Section II, D

  Partial Differential Equations | Part II, 2014

  State the Cauchy-Kovalevskaya theorem, including a definition of the term noncharacteristic.

  For which values of the real number $a$, and for which functions $f$, does the CauchyKovalevskaya theorem ensure that the Cauchy problem

  $$u_{t t}=u_{x x}+a u_{x x x x}, \quad u(x, 0)=0, u_{t}(x, 0)=f(x)$$

  has a local solution?

  Now consider the Cauchy problem (1) in the case that $f(x)=\sum_{m \in \mathbb{Z}} \hat{f}(m) e^{i m x}$ is a smooth $2 \pi$-periodic function.

  (i) Show that if $a \leqslant 0$ there exists a unique smooth solution $u$ for all times, and show that for all $T \geqslant 0$ there exists a number $C=C(T)>0$, independent of $f$, such that

  $$\int_{-\pi}^{+\pi}|u(x, t)|^{2} d x \leqslant C \int_{-\pi}^{+\pi}|f(x)|^{2} d x$$

  for all $t:|t| \leqslant T$.

  (ii) If $a=1$ does there exist a choice of $C=C(T)$ for which (2) holds? Give a full justification for your answer.

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• Paper 2, Section II, D

  Partial Differential Equations | Part II, 2014

  In this question, functions are all real-valued, and

  $$H_{p e r}^{s}=\left\{u=\sum_{m \in \mathbb{Z}} \hat{u}(m) e^{i m x} \in L^{2}:\|u\|_{H^{s}}^{2}=\sum_{m \in \mathbb{Z}}\left(1+m^{2}\right)^{s}|\hat{u}(m)|^{2}<\infty\right\}$$

  are the Sobolev spaces of functions $2 \pi$-periodic in $x$, for $s=0,1,2, \ldots$

  State Parseval's theorem. For $s=0,1$ prove that the norm $\|u\|_{H^{s}}$ is equivalent to the norm ||$_{s}$ defined by

  $$\|u\|_{s}^{2}=\sum_{r=0}^{s} \int_{-\pi}^{+\pi}\left(\partial_{x}^{r} u\right)^{2} d x$$

  Consider the Cauchy problem

  $$u_{t}-u_{x x}=f, \quad u(x, 0)=u_{0}(x), \quad t \geqslant 0,$$

  where $f=f(x, t)$ is a smooth function which is $2 \pi$-periodic in $x$, and the initial value $u_{0}$ is also smooth and $2 \pi$-periodic. Prove that if $u$ is a smooth solution which is $2 \pi$-periodic in $x$, then it satisfies

  $$\int_{0}^{T}\left(u_{t}^{2}+u_{x x}^{2}\right) d t \leqslant C\left(\left\|u_{0}\right\|_{H^{1}}^{2}+\int_{0}^{T} \int_{-\pi}^{\pi}|f(x, t)|^{2} d x d t\right)$$

  for some number $C>0$ which does not depend on $u$ or $f$.

  State the Lax-Milgram lemma. Prove, using the Lax-Milgram lemma, that if

  $$f(x, t)=e^{\lambda t} g(x)$$

  with $g \in H_{p e r}^{0}$ and $\lambda>0$, then there exists a weak solution to (1) of the form $u(x, t)=e^{\lambda t} \phi(x)$ with $\phi \in H_{\text {per. }}^{1}$. Does the same hold for all $\lambda \in \mathbb{R}$ ? Briefly explain your answer.

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• Paper 3, Section II, D

  Partial Differential Equations | Part II, 2014

  (a) Consider variable-coefficient operators of the form

  $$P u=-\sum_{j, k=1}^{n} a_{j k} \partial_{j} \partial_{k} u+\sum_{j=1}^{n} b_{j} \partial_{j} u+c u$$

  whose coefficients are defined on a bounded open set $\Omega \subset \mathbb{R}^{n}$ with smooth boundary $\partial \Omega$. Let $a_{j k}$ satisfy the condition of uniform ellipticity, namely

  $$m\|\xi\|^{2} \leqslant \sum_{j, k=1}^{n} a_{j k}(x) \xi_{j} \xi_{k} \leqslant M\|\xi\|^{2} \quad \text { for all } x \in \Omega \text { and } \xi \in \mathbb{R}^{n}$$

  for suitably chosen positive numbers $m, M$.

  State and prove the weak maximum principle for solutions of $P u=0$. [Any results from linear algebra and calculus needed in your proof should be stated clearly, but need not be proved.]

  (b) Consider the nonlinear elliptic equation

  $$-\Delta u+e^{u}=f$$

  for $u: \mathbb{R}^{n} \rightarrow \mathbb{R}$ satisfying the additional condition

  $$\lim _{|x| \rightarrow \infty} u(x)=0 .$$

  Assume that $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$. Prove that any two $C^{2}$ solutions of (1) which also satisfy (2) are equal.

  Now let $u \in C^{2}\left(\mathbb{R}^{n}\right)$ be a solution of $(1)$ and (2). Prove that if $f(x)<1$ for all $x$ then $u(x)<0$ for all $x$. Prove that if $\max _{x} f(x)=L \geqslant 1$ then $u(x) \leqslant \ln L$ for all $x$.

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• Paper 4, Section II, D

  Partial Differential Equations | Part II, 2014

  (a) Derive the solution of the one-dimensional wave equation

  $$u_{t t}-u_{x x}=0, \quad u(0, x)=u_{0}(x), \quad u_{t}(0, x)=u_{1}(x),$$

  with Cauchy data given by $C^{2}$ functions $u_{j}=u_{j}(x), j=0,1$, and where $x \in \mathbb{R}$ and $u_{t t}=\partial_{t}^{2} u$ etc. Explain what is meant by the property of finite propagation speed for the wave equation. Verify that the solution to (1) satisfies this property.

  (b) Consider the Cauchy problem

  $$u_{t t}-u_{x x}+x^{2} u=0, \quad u(0, x)=u_{0}(x), \quad u_{t}(0, x)=u_{1}(x)$$

  By considering the quantities

  $$e=\frac{1}{2}\left(u_{t}^{2}+u_{x}^{2}+x^{2} u^{2}\right) \quad \text { and } \quad p=-u_{t} u_{x}$$

  prove that solutions of (2) also satisfy the property of finite propagation speed.

  (c) Define what is meant by a strongly continuous one-parameter group of unitary operators on a Hilbert space. Consider the Cauchy problem for the Schrödinger equation for $\psi(x, t) \in \mathbb{C}$ :

  $$i \psi_{t}=-\psi_{x x}+x^{2} \psi, \quad \psi(x, 0)=\psi_{0}(x), \quad-\infty<x<\infty$$

  [In the following you may use without proof the fact that there is an orthonormal set of (real-valued) Schwartz functions $\left\{f_{j}(x)\right\}_{j=1}^{\infty}$ which are eigenfunctions of the differential operator $P=-\partial_{x}^{2}+x^{2}$ with eigenvalues $2 j+1$, i.e.

  $$P f_{j}=(2 j+1) f_{j}, \quad f_{j} \in \mathcal{S}(\mathbb{R}), \quad\left(f_{j}, f_{k}\right)_{L^{2}}=\int_{\mathbb{R}} f_{j}(x) f_{k}(x) d x=\delta_{j k},$$

  and which have the property that any function $u \in L^{2}$ can be written uniquely as a sum $u(x)=\sum_{j}\left(f_{j}, u\right)_{L^{2}} f_{j}(x)$ which converges in the metric defined by the $L^{2}$ norm.]

  Write down the solution to (3) in the case that $\psi_{0}$ is given by a finite sum $\psi_{0}=\sum_{j=1}^{N}\left(f_{j}, \psi_{0}\right)_{L^{2}} f_{j}$ and show that your formula extends to define a strongly continuous one-parameter group of unitary operators on the Hilbert space $L^{2}$ of square-integrable (complex-valued) functions, with inner product $(f, g)_{L^{2}}=\int_{\mathbb{R}} \overline{f(x)} g(x) d x$.

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2013

• Paper 1, Section II, C

  Partial Differential Equations | Part II, 2013

  (i) Discuss briefly the concept of well-posedness of a Cauchy problem for a partial differential equation.

  Solve the Cauchy problem

  $$\partial_{2} u+x_{1} \partial_{1} u=a u^{2}, \quad u\left(x_{1}, 0\right)=\phi\left(x_{1}\right),$$

  where $a \in \mathbb{R}, \phi \in C^{1}(\mathbb{R})$ and $\partial_{i}$ denotes the partial derivative with respect to $x_{i}$ for $i=1,2$.

  For the case $a=0$ show that the solution satisfies $\max _{x_{1} \in \mathbb{R}}\left|u\left(x_{1}, x_{2}\right)\right|=\|\phi\|_{C^{0}}$, where the $C^{r}$ norm on functions $\phi=\phi\left(x_{1}\right)$ of one variable is defined by

  $$\|\phi\|_{C^{r}}=\sum_{i=0}^{r} \max _{x \in \mathbb{R}}\left|\partial_{1}^{i} \phi\left(x_{1}\right)\right|$$

  Deduce that the Cauchy problem is then well-posed in the uniform metric (i.e. the metric determined by the $C^{0}$ norm).

  (ii) State the Cauchy-Kovalevskaya theorem and deduce that the following Cauchy problem for the Laplace equation,

  $$\partial_{1}^{2} u+\partial_{2}^{2} u=0, \quad u\left(x_{1}, 0\right)=0, \partial_{2} u\left(x_{1}, 0\right)=\phi\left(x_{1}\right)$$

  has a unique analytic solution in some neighbourhood of $x_{2}=0$ for any analytic function $\phi=\phi\left(x_{1}\right)$. Write down the solution for the case $\phi\left(x_{1}\right)=\sin \left(n x_{1}\right)$, and hence give a sequence of initial data $\left\{\phi_{n}\left(x_{1}\right)\right\}_{n=1}^{\infty}$ with the property that

  $$\left\|\phi_{n}\right\|_{C^{r}} \rightarrow 0, \quad \text { as } n \rightarrow \infty, \text { for each } r \in \mathbb{N},$$

  whereas $u_{n}$, the corresponding solution of $(*)$, satisfies

  $$\max _{x_{1} \in \mathbb{R}}\left|u_{n}\left(x_{1}, x_{2}\right)\right| \rightarrow+\infty, \quad \text { as } n \rightarrow \infty$$

  for any $x_{2} \neq 0$.

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• Paper 2, Section II, C

  Partial Differential Equations | Part II, 2013

  State the Lax-Milgram lemma.

  Let $\mathbf{V}=\mathbf{V}\left(x_{1}, x_{2}, x_{3}\right)$ be a smooth vector field which is $2 \pi$-periodic in each coordinate $x_{j}$ for $j=1,2,3$. Write down the definition of a weak $H_{p e r}^{1}$ solution for the equation

  $$-\Delta u+\sum_{j} V_{j} \partial_{j} u+u=f$$

  to be solved for $u=u\left(x_{1}, x_{2}, x_{3}\right)$ given $f=f\left(x_{1}, x_{2}, x_{3}\right)$ in $H^{0}$, with both $u$ and $f$ also $2 \pi$-periodic in each co-ordinate. [In this question use the definition

  $$H_{p e r}^{s}=\left\{u=\sum_{m \in \mathbb{Z}^{3}} \hat{u}(m) e^{i m \cdot x} \in L^{2}:\|u\|_{H^{s}}^{2}=\sum_{m \in \mathbb{Z}^{3}}\left(1+\|m\|^{2}\right)^{s}|\hat{u}(m)|^{2}<\infty\right\}$$

  for the Sobolev spaces of functions $2 \pi$-periodic in each coordinate $x_{j}$ and for $\left.s=0,1,2, \ldots\right]$

  If the vector field is divergence-free, prove that there exists a unique weak $H_{p e r}^{1}$ solution for all such $f$.

  Supposing that $\mathbf{V}$ is the constant vector field with components $(1,0,0)$, write down the solution of $(*)$ in terms of Fourier series and show that there exists $C>0$ such that

  $$\|u\|_{H^{2}} \leqslant C\|f\|_{H^{0}}$$

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• Paper 3, Section II, C

  Partial Differential Equations | Part II, 2013

  Define the parabolic boundary $\partial_{p a r} \Omega_{T}$ of the domain $\Omega_{T}=[0,1] \times(0, T]$ for $T>0$.

  Let $u=u(x, t)$ be a smooth real-valued function on $\Omega_{T}$ which satisfies the inequality

  $$u_{t}-a u_{x x}+b u_{x}+c u \leqslant 0$$

  Assume that the coefficients $a, b$ and $c$ are smooth functions and that there exist positive constants $m, M$ such that $m \leqslant a \leqslant M$ everywhere, and $c \geqslant 0$. Prove that

  $$\max _{(x, t) \in \bar{\Omega}_{T}} u(x, t) \leqslant \max _{(x, t) \in \partial_{\text {par }} \Omega_{T}} u^{+}(x, t) .$$

  [Here $u^{+}=\max \{u, 0\}$ is the positive part of the function $u$.]

  Consider a smooth real-valued function $\phi$ on $\Omega_{T}$ such that

  $$\phi_{t}-\phi_{x x}-\left(1-\phi^{2}\right) \phi=0, \quad \phi(x, 0)=f(x)$$

  everywhere, and $\phi(0, t)=1=\phi(1, t)$ for all $t \geqslant 0$. Deduce from $(*)$ that if $f(x) \leqslant 1$ for all $x \in[0,1]$ then $\phi(x, t) \leqslant 1$ for all $(x, t) \in \Omega_{T}$. [Hint: Consider $u=\phi^{2}-1$ and compute $\left.u_{t}-u_{x x} \cdot\right]$

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• Paper 4, Section II, C

  Partial Differential Equations | Part II, 2013

  (i) Show that an arbitrary $C^{2}$ solution of the one-dimensional wave equation $u_{t t}-u_{x x}=0$ can be written in the form $u=F(x-t)+G(x+t)$.

  Hence, deduce the formula for the solution at arbitrary $t>0$ of the Cauchy problem

  $$u_{t t}-u_{x x}=0, \quad u(0, x)=u_{0}(x), \quad u_{t}(0, x)=u_{1}(x)$$

  where $u_{0}, u_{1}$ are arbitrary Schwartz functions.

  Deduce from this formula a theorem on finite propagation speed for the onedimensional wave equation.

  (ii) Define the Fourier transform of a tempered distribution. Compute the Fourier transform of the tempered distribution $T_{t} \in \mathcal{S}^{\prime}(\mathbb{R})$ defined for all $t>0$ by the function

  $$T_{t}(y)= \begin{cases}\frac{1}{2} & \text { if }|y| \leqslant t \\ 0 & \text { if }|y|>t\end{cases}$$

  that is, $\left\langle T_{t}, f\right\rangle=\frac{1}{2} \int_{-t}^{+t} f(y) d y$ for all $f \in \mathcal{S}(\mathbb{R})$. By considering the Fourier transform in $x$, deduce from this the formula for the solution of $(*)$ that you obtained in part (i) in the case $u_{0}=0$.

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2012

• Paper 1, Section II, B

  Partial Differential Equations | Part II, 2012

  Let $u_{0}: \mathbb{R} \rightarrow \mathbb{R}, u_{0} \in C^{1}(\mathbb{R}), u_{0}(x) \geqslant 0$ for all $x \in \mathbb{R}$. Consider the partial differential equation for $u=u(x, y)$,

  $$4 y u_{x}+3 u_{y}=u^{2}, \quad(x, y) \in \mathbb{R}^{2}$$

  subject to the Cauchy condition $u(x, 0)=u_{0}(x)$.

  i) Compute the solution of the Cauchy problem by the method of characteristics.

  ii) Prove that the domain of definition of the solution contains

  $$(x, y) \in \mathbb{R} \times\left(-\infty, \frac{3}{\sup _{x \in \mathbb{R}}\left(u_{0}(x)\right)}\right)$$

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• Paper 2, Section II, B

  Partial Differential Equations | Part II, 2012

  Consider the elliptic Dirichlet problem on $\Omega \subset \mathbb{R}^{n}, \Omega$ bounded with a smooth boundary:

  $$\Delta u-e^{u}=f \text { in } \Omega, \quad u=u_{D} \text { on } \partial \Omega .$$

  Assume that $u_{D} \in L^{\infty}(\partial \Omega)$ and $f \in L^{\infty}(\Omega)$.

  (i) State the strong Minimum-Maximum Principle for uniformly elliptic operators.

  (ii) Prove that there exists at most one classical solution of the boundary value problem.

  (iii) Assuming further that $f \geqslant 0$ in $\Omega$, use the maximum principle to obtain an upper bound on the solution (assuming that it exists).

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• Paper 3, Section II, 30B

  Partial Differential Equations | Part II, 2012

  Consider the nonlinear partial differential equation for a function $u(x, t), x \in \mathbb{R}^{n}, t>0$,

  $$\begin{array}{r} u_{t}=\Delta u-\alpha|\nabla u|^{2} \\ \text { subject to } \quad u(x, 0)=u_{0}(x) \end{array}$$

  where $u_{0} \in L^{\infty}\left(\mathbb{R}^{n}\right)$.

  (i) Find a transformation $w:=F(u)$ such that $w$ satisfies the heat equation

  $$w_{t}=\Delta w, \quad x \in \mathbb{R}^{n}$$

  if (1) holds for $u$.

  (ii) Use the transformation obtained in (i) (and its inverse) to find a solution to the initial value problem (1), (2).

  [Hint. Use the fundamental solution of the heat equation.]

  (iii) The equation (1) is posed on a bounded domain $\Omega \subseteq \mathbb{R}^{n}$ with smooth boundary, subject to the initial condition (2) on $\Omega$ and inhomogeneous Dirichlet boundary conditions

  $$u=u_{D} \text { on } \partial \Omega$$

  where $u_{D}$ is a bounded function. Use the maximum-minimum principle to prove that there exists at most one classical solution of this boundary value problem.

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• Paper 4, Section II, 30B

  Partial Differential Equations | Part II, 2012

  i) State the Lax-Milgram lemma.

  ii) Consider the boundary value problem

  $$\begin{aligned} \Delta^{2} u-\Delta u+u=f & \text { in } \Omega \\ u=\nabla u \cdot \gamma=0 & \text { on } \partial \Omega \end{aligned}$$

  where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with a smooth boundary, $\gamma$ is the exterior unit normal vector to $\partial \Omega$, and $f \in L^{2}(\Omega)$. Show (using the Lax-Milgram lemma) that the boundary value problem has a unique weak solution in the space

  $$H_{0}^{2}(\Omega):=\{u: \Omega \rightarrow \mathbb{R} ; u=\nabla u \cdot \gamma=0 \text { on } \partial \Omega\}$$

  [Hint. Show that

  $$\|\Delta u\|_{L^{2}(\Omega)}^{2}=\sum_{i, j=1}^{n}\left\|\frac{\partial^{2} u}{\partial x_{i} \partial x_{j}}\right\|_{L^{2}(\Omega)}^{2} \quad \text { for all } u \in C_{0}^{\infty}(\Omega)$$

  and then use the fact that $C_{0}^{\infty}(\Omega)$ is dense in $\left.H_{0}^{2}(\Omega) .\right]$

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2011

• Paper 1, Section II, A

  Partial Differential Equations | Part II, 2011

  Let $H=H(x, v), x, v \in \mathbb{R}^{n}$, be a smooth real-valued function which maps $\mathbb{R}^{2 n}$ into $\mathbb{R}$. Consider the initial value problem for the equation

  $$\begin{aligned} &f_{t}+\nabla_{v} H \cdot \nabla_{x} f-\nabla_{x} H \cdot \nabla_{v} f=0, \quad x, v \in \mathbb{R}^{n}, t>0 \\ &f(x, v, t=0)=f_{I}(x, v), \quad x, v \in \mathbb{R}^{n} \end{aligned}$$

  for the unknown function $f=f(x, v, t)$.

  (i) Use the method of characteristics to solve the initial value problem, locally in time.

  (ii) Let $f_{I} \geqslant 0$ on $\mathbb{R}^{2 n}$. Use the method of characteristics to prove that $f$ remains non-negative (as long as it exists).

  (iii) Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be smooth. Prove that

  $$\int_{\mathbb{R}^{2 n}} F(f(x, v, t)) d x d v=\int_{\mathbb{R}^{2 n}} F\left(f_{I}(x, v)\right) d x d v$$

  as long as the solution exists.

  (iv) Let $H$ be independent of $x$, namely $H(x, v)=a(v)$, where $a$ is smooth and realvalued. Give the explicit solution of the initial value problem.

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• Paper 2, Section II, A

  Partial Differential Equations | Part II, 2011

  Consider the Schrödinger equation

  $$\begin{aligned} i \partial_{t} \psi(t, x) &=-\frac{1}{2} \Delta \psi(t, x)+V(x) \psi(t, x), \quad x \in \mathbb{R}^{n}, t>0 \\ \psi(t=0, x) &=\psi_{I}(x), \quad x \in \mathbb{R}^{n} \end{aligned}$$

  where $V$ is a smooth real-valued function.

  Prove that, for smooth solutions, the following equations are valid for all $t>0$ :

  (i)

  $$\int_{\mathbb{R}^{n}}|\psi(t, x)|^{2} d x=\int_{\mathbb{R}^{n}}\left|\psi_{I}(x)\right|^{2} d x$$

  (ii)

  $$\begin{aligned} &\int_{\mathbb{R}^{n}} \frac{1}{2}|\nabla \psi(t, x)|^{2} d x+\int_{\mathbb{R}^{n}} V(x)|\psi(t, x)|^{2} d x \\ &=\int_{\mathbb{R}^{n}} \frac{1}{2}\left|\nabla \psi_{I}(x)\right|^{2} d x+\int_{\mathbb{R}^{n}} V(x)\left|\psi_{I}(x)\right|^{2} d x \end{aligned}$$

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• Paper 3, Section II, A

  Partial Differential Equations | Part II, 2011

  (a) State the local existence theorem of a classical solution of the Cauchy problem

  $$\begin{aligned} &a\left(x_{1}, x_{2}, u\right) \frac{\partial u}{\partial x_{1}}+b\left(x_{1}, x_{2}, u\right) \frac{\partial u}{\partial x_{2}}=c\left(x_{1}, x_{2}, u\right) \\ &\left.u\right|_{\Gamma}=u_{0} \end{aligned}$$

  where $\Gamma$ is a smooth curve in $\mathbb{R}^{2}$.

  (b) Solve, by using the method of characteristics,

  $$\begin{aligned} &2 x_{1} \frac{\partial u}{\partial x_{1}}+4 x_{2} \frac{\partial u}{\partial x_{2}}=u^{2} \\ &u\left(x_{1}, 2\right)=h \end{aligned}$$

  where $h>0$ is a constant. What is the maximal domain of existence in which $u$ is a solution of the Cauchy problem?

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• Paper 4, Section II, A

  Partial Differential Equations | Part II, 2011

  Consider the functional

  $$E(u)=\frac{1}{2} \int_{\Omega}|\nabla u|^{2} d x+\int_{\Omega} F(u, x) d x$$

  where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary and $F: \mathbb{R} \times \Omega \rightarrow \mathbb{R}$ is smooth. Assume that $F(u, x)$ is convex in $u$ for all $x \in \Omega$ and that there is a $K>0$ such that

  $$-K \leqslant F(v, x) \leqslant K\left(|v|^{2}+1\right) \quad \forall v \in \mathbb{R}, x \in \Omega$$

  (i) Prove that $E$ is well-defined on $H_{0}^{1}(\Omega)$, bounded from below and strictly convex. Assume without proof that $E$ is weakly lower-semicontinuous. State this property. Conclude the existence of a unique minimizer of $E$.

  (ii) Which elliptic boundary value problem does the minimizer solve?

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2010

• Paper 1, Section II, E

  Partial Differential Equations | Part II, 2010

  (a) Solve by using the method of characteristics

  $$x_{1} \frac{\partial}{\partial x_{1}} u+2 x_{2} \frac{\partial}{\partial x_{2}} u=5 u, \quad u\left(x_{1}, 1\right)=g\left(x_{1}\right),$$

  where $g: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. What is the maximal domain in $\mathbb{R}^{2}$ in which $u$ is a solution of the Cauchy problem?

  (b) Prove that the function

  $$u(x, t)=\left\{\begin{array}{cl} 0, & x<0, t>0 \\ x / t, & 0<x<t, t>0 \\ 1, & x>t>0 \end{array}\right.$$

  is a weak solution of the Burgers equation

  $$\frac{\partial}{\partial t} u+\frac{1}{2} \frac{\partial}{\partial x} u^{2}=0, \quad x \in \mathbb{R}, t>0$$

  with initial data

  $$u(x, 0)= \begin{cases}0, & x<0 \\ 1, & x>0\end{cases}$$

  (c) Let $u=u(x, t), x \in \mathbb{R}, t>0$ be a piecewise $C^{1}$-function with a jump discontinuity along the curve

  $$\Gamma: x=s(t)$$

  and let $u$ solve the Burgers equation $(*)$ on both sides of $\Gamma$. Prove that $u$ is a weak solution of (1) if and only if

  $$\dot{s}(t)=\frac{1}{2}\left(u_{l}(t)+u_{r}(t)\right)$$

  holds, where $u_{l}(t), u_{r}(t)$ are the one-sided limits

  $$u_{l}(t)=\lim _{x \nearrow_{s}(t)^{-}} u(x, t), \quad u_{r}(t)=\lim _{x \searrow s(t)^{+}} u(x, t)$$

  [Hint: Multiply the equation by a test function $\phi \in C_{0}^{\infty}(\mathbb{R} \times[0, \infty))$, split the integral appropriately and integrate by parts. Consider how the unit normal vector along $\Gamma$ can be expressed in terms of $\dot{s}$.]

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• Paper 2, Section II, E

  Partial Differential Equations | Part II, 2010

  (a) State the Lax-Milgram lemma. Use it to prove that there exists a unique function $u$ in the space

  $$H_{\partial}^{2}(\Omega)=\left\{u \in H^{2}(\Omega) ;\left.u\right|_{\partial \Omega}=\partial u /\left.\partial \gamma\right|_{\partial \Omega}=0\right\}$$

  where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary and $\gamma$ its outwards unit normal vector, which is the weak solution of the equations

  $$\begin{aligned} \Delta^{2} u &=f \text { in } \Omega, \\ u &=\frac{\partial u}{\partial \gamma}=0 \text { on } \partial \Omega, \end{aligned}$$

  for $f \in L^{2}(\Omega), \Delta$ the Laplacian and $\Delta^{2}=\Delta \Delta$.

  [Hint: Use regularity of the solution of the Dirichlet problem for the Poisson equation.]

  (b) Let $\Omega \subset \mathbb{R}^{n}$ be a bounded domain with smooth boundary. Let $u \in H^{1}(\Omega)$ and denote

  $$\bar{u}=\int_{\Omega} u d^{n} x / \int_{\Omega} d^{n} x$$

  The following Poincaré-type inequality is known to hold

  $$\|u-\bar{u}\|_{L^{2}} \leqslant C\|\nabla u\|_{L^{2}}$$

  where $C$ only depends on $\Omega$. Use the Lax-Milgram lemma and this Poincaré-type inequality to prove that the Neumann problem

  $$\begin{aligned} &\Delta u=f \text { in } \Omega \\ &\frac{\partial u}{\partial \gamma}=0 \text { on } \partial \Omega \end{aligned}$$

  has a unique weak solution in the space

  $$H_{-}^{1}(\Omega)=H^{1}(\Omega) \cap\{u: \Omega \rightarrow \mathbb{R} ; \bar{u}=0\}$$

  if and only if $\bar{f}=0$.

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• Paper 3, Section II, 30E

  Partial Differential Equations | Part II, 2010

  Consider the Schrödinger equation

  $$i \partial_{t} \Psi=-\frac{1}{2} \Delta \Psi, \quad x \in \mathbb{R}^{n}, t>0$$

  for complex-valued solutions $\Psi(x, t)$ and where $\Delta$ is the Laplacian.

  (a) Derive, by using a Fourier transform and its inversion, the fundamental solution of the Schrödinger equation. Obtain the solution of the initial value problem

  $$\begin{aligned} i \partial_{t} \Psi=-\frac{1}{2} \Delta \Psi, & x \in \mathbb{R}^{n}, \quad t>0 \\ \Psi(x, 0)=f(x), & x \in \mathbb{R}^{n} \end{aligned}$$

  as a convolution.

  (b) Consider the Wigner-transform of the solution of the Schrödinger equation

  $$w(x, \xi, t)=\frac{1}{(2 \pi)^{n}} \int_{\mathbb{R}^{n}} \Psi\left(x+\frac{1}{2} y, t\right) \bar{\Psi}\left(x-\frac{1}{2} y, t\right) e^{-i y \cdot \xi} \mathrm{d}^{n} y$$

  defined for $x \in \mathbb{R}^{n}, \xi \in \mathbb{R}^{n}, t>0$. Derive an evolution equation for $w$ by using the Schrödinger equation. Write down the solution of this evolution equation for given initial data $w(x, \xi, 0)=g(x, \xi)$.

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• Paper 4, Section II, 30E

  Partial Differential Equations | Part II, 2010

  a) Solve the Dirichlet problem for the Laplace equation in a disc in $\mathbb{R}^{2}$

  $$\begin{aligned} \Delta u &=0 \quad \text { in } \quad G=\left\{x^{2}+y^{2}<R^{2}\right\} \subseteq \mathbb{R}^{2}, R>0 \\ u &=u_{D} \quad \text { on } \quad \partial G \end{aligned}$$

  using polar coordinates $(r, \varphi)$ and separation of variables, $u(x, y)=R(r) \Theta(\varphi)$. Then use the ansatz $R(r)=r^{\alpha}$ for the radial function.

  b) Solve the Dirichlet problem for the Laplace equation in a square in $\mathbb{R}^{2}$

  $$\begin{aligned} &\Delta u=0 \quad \text { in } \quad G=[0, a] \times[0, a] \\ &u(x, 0)=f_{1}(x), \quad u(x, a)=f_{2}(x), \quad u(0, y)=f_{3}(y), \quad u(a, y)=f_{4}(y) \end{aligned}$$

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2009

• Paper 1, Section II, B

  Partial Differential Equations | Part II, 2009

  Consider the initial value problem for the so-called Liouville equation

  $$\begin{gathered} f_{t}+v \cdot \nabla_{x} f-\nabla V(x) \cdot \nabla_{v} f=0,(x, v) \in \mathbb{R}^{2 d}, t \in \mathbb{R} \\ f(x, v, t=0)=f_{I}(x, v) \end{gathered}$$

  for the function $f=f(x, v, t)$ on $\mathbb{R}^{2 d} \times \mathbb{R}$. Assume that $V=V(x)$ is a given function with $V, \nabla_{x} V$ Lipschitz continuous on $\mathbb{R}^{d}$.

  (i) Let $f_{I}(x, v)=\delta\left(x-x_{0}, v-v_{0}\right)$, for $x_{0}, v_{0} \in \mathbb{R}^{d}$ given. Show that a solution $f$ is given by

  $$f(x, v, t)=\delta\left(x-\hat{x}\left(t, x_{0}, v_{0}\right), v-\hat{v}\left(t, x_{0}, v_{0}\right)\right)$$

  where $(\hat{x}, \hat{v})$ solve the Newtonian system

  $$\begin{array}{ll} \dot{\hat{x}}=\hat{v}, & \hat{x}(t=0)=x_{0} \\ \dot{\hat{v}}=-\nabla V(\hat{x}), & \hat{v}(t=0)=v_{0} \end{array}$$

  (ii) Let $f_{I} \in L_{l o c}^{1}\left(\mathbb{R}^{2 d}\right), f_{I} \geqslant 0$. Prove (by using characteristics) that $f$ remains nonnegative (as long as it exists).

  (iii) Let $f_{I} \in L^{p}\left(\mathbb{R}^{2 d}\right), f_{I} \geqslant 0$ on $\mathbb{R}^{2 d}$. Show (by a formal argument) that

  $$\|f(\cdot, \cdot, t)\|_{L^{p}\left(\mathbb{R}^{2 d}\right)}=\left\|f_{I}\right\|_{L^{p}\left(\mathbb{R}^{2 d}\right)}$$

  for all $t \in \mathbb{R}, 1 \leqslant p<\infty$.

  (iv) Let $V(x)=\frac{1}{2}|x|^{2}$. Use the method of characteristics to solve the initial value problem for general initial data.

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• Paper 2, Section II, B

  Partial Differential Equations | Part II, 2009

  (a) Solve the initial value problem for the Burgers equation

  $$\begin{gathered} u_{t}+\frac{1}{2}\left(u^{2}\right)_{x}=0, \quad x \in \mathbb{R}, t>0 \\ u(x, t=0)=u_{I}(x) \end{gathered}$$

  where

  $$u_{I}(x)= \begin{cases}1, & x<0 \\ 1-x, & 0<x<1 \\ 0, & x>1\end{cases}$$

  Use the method of characteristics. What is the maximal time interval in which this (weak) solution is well defined? What is the regularity of this solution?

  (b) Apply the method of characteristics to the Burgers equation subject to the initial condition

  $$u_{I}(x)= \begin{cases}1, & x>0 \\ 0, & x<0\end{cases}$$

  In $\{(x, t) \mid 0<x<t\}$ use the ansatz $u(x, t)=f\left(\frac{x}{t}\right)$ and determine $f$.

  (c) Using the method of characteristics show that the initial value problem for the Burgers equation has a classical solution defined for all $t>0$ if $u_{I}$ is continuously differentiable and

  $$\frac{d u_{I}}{d x}(x)>0$$

  for all $x \in \mathbb{R}$.

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• Paper 3, Section II, B

  Partial Differential Equations | Part II, 2009

  (a) Consider the nonlinear elliptic problem

  $$\begin{cases}\Delta u=f(u, x), & x \in \Omega \subseteq \mathbb{R}^{d} \\ u=u_{D}, & x \in \partial \Omega\end{cases}$$

  Let $\frac{\partial f}{\partial u}(y, x) \geqslant 0$ for all $y \in \mathbb{R}, x \in \Omega$. Prove that there exists at most one classical solution.

  [Hint: Use the weak maximum principle.]

  (b) Let $\varphi \in \mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ be a radial function. Prove that the Fourier transform of $\varphi$ is radial too.

  (c) Let $\varphi \in \mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ be a radial function. Solve

  $$-\Delta u+u=\varphi(x), \quad x \in \mathbb{R}^{n}$$

  by Fourier transformation and prove that $u$ is a radial function.

  (d) State the Lax-Milgram lemma and explain its use in proving the existence and uniqueness of a weak solution of

  $$\begin{gathered} -\Delta u+a(x) u=f(x), x \in \Omega \\ u=0 \text { on } \partial \Omega \end{gathered}$$

  where $\Omega \subseteq \mathbb{R}^{d}$ bounded, $0 \leqslant \underline{a} \leqslant a(x) \leqslant \bar{a}<\infty$ for all $x \in \Omega$ and $f \in L^{2}(\Omega)$.

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• Paper 4, Section II, B

  Partial Differential Equations | Part II, 2009

  Consider the two-dimensional domain

  $$G=\left\{(x, y) \mid R_{1}^{2}<x^{2}+y^{2}<R_{2}^{2}\right\}$$

  where $0<R_{1}<R_{2}<\infty$. Solve the Dirichlet boundary value problem for the Laplace equation

  $$\begin{gathered} \Delta u=0 \text { in } G, \\ u=u_{1}(\varphi), r=R_{1}, \\ u=u_{2}(\varphi), r=R_{2}, \end{gathered}$$

  where $(r, \varphi)$ are polar coordinates. Assume that $u_{1}, u_{2}$ are $2 \pi$-periodic functions on the real line and $u_{1}, u_{2} \in L_{l o c}^{2}(\mathbb{R})$.

  [Hint: Use separation of variables in polar coordinates, $u=R(r) \Phi(\varphi)$, with periodic boundary conditions for the function $\Phi$ of the angle variable. Use an ansatz of the form $R(r)=r^{\alpha}$ for the radial function.]

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2008

• 1.II.29C

  Partial Differential Equations | Part II, 2008

  (i) State the local existence theorem for the first order quasi-linear partial differential equation

  $$\sum_{j=1}^{n} a_{j}(x, u) \frac{\partial u}{\partial x_{j}}=b(x, u)$$

  which is to be solved for a real-valued function with data specified on a hypersurface $S$. Include a definition of "non-characteristic" in your answer.

  (ii) Consider the linear constant-coefficient case (that is, when all the functions $a_{1}, \ldots, a_{n}$ are real constants and $b(x, u)=c x+d$ for some $c=\left(c_{1}, \ldots, c_{n}\right)$ with $c_{1}, \ldots, c_{n}$ real and $d$ real) and with the hypersurface $S$ taken to be the hyperplane $\mathbf{x} \cdot \mathbf{n}=0$. Explain carefully the relevance of the non-characteristic condition in obtaining a solution via the method of characteristics.

  (iii) Solve the equation

  $$\frac{\partial u}{\partial y}+u \frac{\partial u}{\partial x}=0$$

  with initial data $u(0, y)=-y$ prescribed on $x=0$, for a real-valued function $u(x, y)$. Describe the domain on which your solution is $C^{1}$ and comment on this in relation to the theorem stated in (i).

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• 2.II.30C

  Partial Differential Equations | Part II, 2008

  (i) Define the concept of "fundamental solution" of a linear constant-coefficient partial differential operator and write down the fundamental solution for the operator $-\Delta$ on $\mathbb{R}^{3}$.

  (ii) State and prove the mean value property for harmonic functions on $\mathbb{R}^{3}$.

  (iii) Let $u \in C^{2}\left(\mathbb{R}^{3}\right)$ be a harmonic function which satisfies $u(p) \geqslant 0$ at every point $p$ in an open set $\Omega \subset \mathbb{R}^{3}$. Show that if $B(z, r) \subset B(w, R) \subset \Omega$, then

  $$u(w) \geqslant\left(\frac{r}{R}\right)^{3} u(z) .$$

  Assume that $B(x, 4 r) \subset \Omega$. Deduce, by choosing $R=3 r$ and $w, z$ appropriately, that

  $$\inf _{B(x, r)} u \geqslant 3^{-3} \sup _{B(x, r)} u .$$

  [In (iii), $B(z, \rho)=\left\{x \in \mathbb{R}^{3}:\|x-z\|<\rho\right\}$ is the ball of radius $\rho>0$ centred at $\left.z \in \mathbb{R}^{3} \cdot\right]$

  comment

• 3.II.29C

  Partial Differential Equations | Part II, 2008

  Let $C_{p e r}^{\infty}=\left\{u \in C^{\infty}(\mathbb{R}): u(x+2 \pi)=u(x)\right\}$ be the space of smooth $2 \pi$-periodic functions of one variable.

  (i) For $f \in C_{p e r}^{\infty}$ show that there exists a unique $u_{f} \in C_{p e r}^{\infty}$ such that

  $$-\frac{d^{2} u_{f}}{d x^{2}}+u_{f}=f$$

  (ii) Show that $I_{f}\left[u_{f}+\phi\right]>I_{f}\left[u_{f}\right]$ for every $\phi \in C_{p e r}^{\infty}$ which is not identically zero, where $I_{f}: C_{p e r}^{\infty} \rightarrow \mathbb{R}$ is defined by

  $$I_{f}[u]=\frac{1}{2} \int_{-\pi}^{+\pi}\left[\left(\frac{\partial u}{\partial x}\right)^{2}+u^{2}-2 f(x) u\right] d x$$

  (iii) Show that the equation

  $$\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}+u=f(x)$$

  with initial data $u(0, x)=u_{0}(x) \in C_{\text {per }}^{\infty}$ has, for $t>0$, a smooth solution $u(t, x)$ such that $u(t, \cdot) \in C_{p e r}^{\infty}$ for each fixed $t>0$. Give a representation of this solution as a Fourier series in $x$. Calculate $\lim _{t \rightarrow+\infty} u(t, x)$ and comment on your answer in relation to (i).

  (iv) Show that $I_{f}[u(t, \cdot)] \leqslant I_{f}[u(s, \cdot)]$ for $t>s>0$, and that $I_{f}[u(t, \cdot)] \rightarrow I_{f}\left[u_{f}\right]$ as $t \rightarrow+\infty$.

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• 4.II.30C

  Partial Differential Equations | Part II, 2008

  (i) Define the Fourier transform $\hat{f}=\mathcal{F}(f)$ of a Schwartz function $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$, and also of a tempered distribution $u \in \mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$.

  (ii) From your definition, compute the Fourier transform of the distribution $W_{t} \in \mathcal{S}^{\prime}\left(\mathbb{R}^{3}\right)$ given by

  $$W_{t}(\psi)=<W_{t}, \psi>=\frac{1}{4 \pi t} \int_{\|y\|=t} \psi(y) d \Sigma(y)$$

  for every Schwartz function $\psi \in \mathcal{S}\left(\mathbb{R}^{3}\right)$. Here $d \Sigma(y)=t^{2} d \Omega(y)$ is the integration element on the sphere of radius $t$.

  Hence deduce the formula of Kirchoff for the solution of the initial value problem for the wave equation in three space dimensions,

  $$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=0$$

  with initial data $u(0, x)=0$ and $\frac{\partial u}{\partial t}(0, x)=g(x), x \in \mathbb{R}^{3}$, where $g \in \mathcal{S}\left(\mathbb{R}^{3}\right)$. Explain briefly why the formula is also valid for arbitrary smooth $g \in C^{\infty}\left(\mathbb{R}^{3}\right)$.

  (iii) Show that any $C^{2}$ solution of the initial value problem in (ii) is given by the formula derived in (ii) (uniqueness).

  (iv) Show that any two $C^{2}$ solutions of the initial value problem for

  $$\frac{\partial^{2} u}{\partial t^{2}}+\frac{\partial u}{\partial t}-\Delta u=0$$

  with the same initial data as in (ii), also agree for any $t>0$.

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2007

• 1.II.29A

  Partial Differential Equations | Part II, 2007

  (i) Consider the problem of solving the equation

  $$\sum_{j=1}^{n} a_{j}(\mathbf{x}) \frac{\partial u}{\partial x_{j}}=b(\mathbf{x}, u)$$

  for a $C^{1}$ function $u=u(\mathbf{x})=u\left(x_{1}, \ldots, x_{n}\right)$, with data specified on a $C^{1}$ hypersurface $\mathcal{S} \subset \mathbb{R}^{n}$

  $$u(\mathbf{x})=\phi(\mathbf{x}), \quad \forall \mathbf{x} \in \mathcal{S}$$

  Assume that $a_{1}, \ldots, a_{n}, \phi, b$ are $C^{1}$ functions. Define the characteristic curves and explain what it means for the non-characteristic condition to hold at a point on $\mathcal{S}$. State a local existence and uniqueness theorem for the problem.

  (ii) Consider the case $n=2$ and the equation

  $$\frac{\partial u}{\partial x_{1}}-\frac{\partial u}{\partial x_{2}}=x_{2} u$$

  with data $u\left(x_{1}, 0\right)=\phi\left(x_{1}, 0\right)=f\left(x_{1}\right)$ specified on the axis $\left\{\mathbf{x} \in \mathbb{R}^{2}: x_{2}=0\right\}$. Obtain a formula for the solution.

  (iii) Consider next the case $n=2$ and the equation

  $$\frac{\partial u}{\partial x_{1}}-\frac{\partial u}{\partial x_{2}}=0$$

  with data $u(\mathbf{g}(s))=\phi(\mathbf{g}(s))=f(s)$ specified on the hypersurface $\mathcal{S}$, which is given parametrically as $\mathcal{S} \equiv\left\{\mathbf{x} \in \mathbb{R}^{2}: \mathbf{x}=\mathbf{g}(s)\right\}$ where $\mathbf{g}: \mathbb{R} \rightarrow \mathbb{R}^{2}$ is defined by

  $$\begin{aligned} &\mathbf{g}(s)=(s, 0), \quad s<0 \\ &\mathbf{g}(s)=\left(s, s^{2}\right), \quad s \geqslant 0 \end{aligned}$$

  Find the solution $u$ and show that it is a global solution. (Here "global" means $u$ is $C^{1}$ on all of $\mathbb{R}^{2}$.)

  (iv) Consider next the equation

  $$\frac{\partial u}{\partial x_{1}}+\frac{\partial u}{\partial x_{2}}=0$$

  to be solved with the same data given on the same hypersurface as in (iii). Explain, with reference to the characteristic curves, why there is generally no global $C^{1}$ solution. Discuss the existence of local solutions defined in some neighbourhood of a given point $\mathbf{y} \in \mathcal{S}$ for various $\mathbf{y}$. [You need not give formulae for the solutions.]

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• 2.II.30A

  Partial Differential Equations | Part II, 2007

  Define (i) the Fourier transform of a tempered distribution $T \in \mathcal{S}^{\prime}\left(\mathbb{R}^{3}\right)$, and (ii) the convolution $T * g$ of a tempered distribution $T \in \mathcal{S}^{\prime}\left(\mathbb{R}^{3}\right)$ and a Schwartz function $g \in \mathcal{S}\left(\mathbb{R}^{3}\right)$. Give a formula for the Fourier transform of $T * g$ ("convolution theorem").

  Let $t>0$. Compute the Fourier transform of the tempered distribution $A_{t} \in \mathcal{S}^{\prime}\left(\mathbb{R}^{3}\right)$ defined by

  $$\left\langle A_{t}, \phi\right\rangle=\int_{\|y\|=t} \phi(y) d \Sigma(y), \quad \forall \phi \in \mathcal{S}\left(\mathbb{R}^{3}\right),$$

  and deduce the Kirchhoff formula for the solution $u(t, x)$ of

  $$\begin{gathered} \frac{\partial^{2} u}{\partial t^{2}}-\Delta u=0, \\ u(0, x)=0, \quad \frac{\partial u}{\partial t}(0, x)=g(x), \quad g \in \mathcal{S}\left(\mathbb{R}^{3}\right) \end{gathered}$$

  Prove, by consideration of the quantities $e=\frac{1}{2}\left(u_{t}^{2}+|\nabla u|^{2}\right)$ and $p=-u_{t} \nabla u$, that any $C^{2}$ solution is also given by the Kirchhoff formula (uniqueness).

  Prove a corresponding uniqueness statement for the initial value problem

  $$\begin{gathered} \frac{\partial^{2} w}{\partial t^{2}}-\Delta w+V(x) w=0 \\ w(0, x)=0, \quad \frac{\partial w}{\partial t}(0, x)=g(x), \quad g \in \mathcal{S}\left(\mathbb{R}^{3}\right) \end{gathered}$$

  where $V$ is a smooth positive real-valued function of $x \in \mathbb{R}^{3}$ only.

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• 3.II.29A

  Partial Differential Equations | Part II, 2007

  Write down the formula for the solution $u=u(t, x)$ for $t>0$ of the initial value problem for the heat equation in one space dimension

  $$\begin{aligned} &\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=0 \\ &u(0, x)=g(x) \end{aligned}$$

  for $g: \mathbb{R} \rightarrow \mathbb{C}$ a given smooth bounded function.

  Define the distributional derivative of a tempered distribution $T \in \mathcal{S}^{\prime}(\mathbb{R})$. Define a fundamental solution of a constant-coefficient linear differential operator $P$, and show that the distribution defined by the function $\frac{1}{2} e^{-|x|}$ is a fundamental solution for the operator $P=-\frac{d^{2}}{d x^{2}}+1$.

  For the equation

  $$\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=e^{t} \phi(x)$$

  where $\phi \in \mathcal{S}(\mathbb{R})$, prove that there is a unique solution of the form $e^{t} v(x)$ with $v \in \mathcal{S}(\mathbb{R})$. Hence write down the solution of $(*)$ with general initial data $u(0, x)=f(x)$ and describe the large time behaviour.

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• 4.II.30A

  Partial Differential Equations | Part II, 2007

  State and prove the mean value property for harmonic functions on $\mathbb{R}^{3}$.

  Obtain a generalization of the mean value property for sub-harmonic functions on $\mathbb{R}^{3}$, i.e. $C^{2}$ functions for which

  $$-\Delta u(x) \leqslant 0$$

  for all $x \in \mathbb{R}^{3}$.

  Let $\phi \in C^{2}\left(\mathbb{R}^{3} ; \mathbb{C}\right)$ solve the equation

  $$-\Delta \phi+i V(x) \phi=0$$

  where $V$ is a real-valued continuous function. By considering the function $w(x)=|\phi(x)|^{2}$ show that, on any ball $B(y, R)=\{x:\|x-y\|<R\} \subset \mathbb{R}^{3}$,

  $$\sup _{x \in B(y, R)}|\phi(x)| \leqslant \sup _{\|x-y\|=R}|\phi(x)| .$$

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2006

• 1.II.29A

  Partial Differential Equations | Part II, 2006

  (a) State a local existence theorem for solving first order quasi-linear partial differential equations with data specified on a smooth hypersurface.

  (b) Solve the equation

  $$\frac{\partial u}{\partial x}+x \frac{\partial u}{\partial y}=0$$

  with boundary condition $u(x, 0)=f(x)$ where $f \in C^{1}(\mathbb{R})$, making clear the domain on which your solution is $C^{1}$. Comment on this domain with reference to the noncharacteristic condition for an initial hypersurface (including a definition of this concept).

  (c) Solve the equation

  $$u^{2} \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$$

  with boundary condition $u(x, 0)=x$ and show that your solution is $C^{1}$ on some open set containing the initial hypersurface $y=0$. Comment on the significance of this, again with reference to the non-characteristic condition.

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• 2.II.30A

  Partial Differential Equations | Part II, 2006

  Define a fundamental solution of a constant-coefficient linear partial differential operator, and prove that the distribution defined by the function $N: \mathbb{R}^{3} \rightarrow \mathbb{R}$

  $$N(x)=(4 \pi|x|)^{-1}$$

  is a fundamental solution of the operator $-\Delta$ on $\mathbb{R}^{3}$.

  State and prove the mean value property for harmonic functions on $\mathbb{R}^{3}$ and deduce that any two smooth solutions of

  $$-\Delta u=f, \quad f \in C^{\infty}\left(\mathbb{R}^{3}\right)$$

  which satisfy the condition

  $$\lim _{|x| \rightarrow \infty} u(x)=0$$

  are in fact equal

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• 3.II.29A

  Partial Differential Equations | Part II, 2006

  Write down the formula for the solution $u=u(t, x)$ for $t>0$ of the initial value problem for the $n$-dimensional heat equation

  $$\begin{gathered} \frac{\partial u}{\partial t}-\Delta u=0 \\ u(0, x)=g(x) \end{gathered}$$

  for $g: \mathbb{R}^{n} \rightarrow \mathbb{C}$ a given smooth bounded function.

  State and prove the Duhamel principle giving the solution $v(t, x)$ for $t>0$ to the inhomogeneous initial value problem

  $$\begin{aligned} &\frac{\partial v}{\partial t}-\Delta v=f \\ &v(0, x)=g(x) \end{aligned}$$

  for $f=f(t, x)$ a given smooth bounded function.

  For the case $n=4$ and when $f=f(x)$ is a fixed Schwartz function (independent of $t)$, find $v(t, x)$ and show that $w(x)=\lim _{t \rightarrow+\infty} v(t, x)$ is a solution of

  $$-\Delta w=f \text {. }$$

  [Hint: you may use without proof the fact that the fundamental solution of the Laplacian on $\mathbb{R}^{4}$ is $\left.-1 /\left(4 \pi^{2}|x|^{2}\right) .\right]$

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• 4.II.30A

  Partial Differential Equations | Part II, 2006

  (a) State the Fourier inversion theorem for Schwartz functions $\mathcal{S}(\mathbb{R})$ on the real line. Define the Fourier transform of a tempered distribution and compute the Fourier transform of the distribution defined by the function $F(x)=\frac{1}{2}$ for $-t \leqslant x \leqslant+t$ and $F(x)=0$ otherwise. (Here $t$ is any positive number.)

  Use the Fourier transform in the $x$ variable to deduce a formula for the solution to the one dimensional wave equation

  $$u_{t t}-u_{x x}=0, \quad \text { with initial data } \quad u(0, x)=0, \quad u_{t}(0, x)=g(x),$$

  for $g$ a Schwartz function. Explain what is meant by "finite propagation speed" and briefly explain why the formula you have derived is in fact valid for arbitrary smooth $g \in C^{\infty}(\mathbb{R})$.

  (b) State a theorem on the representation of a smooth $2 \pi$-periodic function $g$ as a Fourier series

  $$g(x)=\sum_{\alpha \in \mathbb{Z}} \hat{g}(\alpha) e^{i \alpha x}$$

  and derive a representation for solutions to $(*)$ as Fourier series in $x$.

  (c) Verify that the formulae obtained in (a) and (b) agree for the case of smooth $2 \pi$ periodic $g$.

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2005

• 1.II.29C

  Partial Differential Equations | Part II, 2005

  Consider the equation

  $$x_{2} \frac{\partial u}{\partial x_{1}}-x_{1} \frac{\partial u}{\partial x_{2}}+a \frac{\partial u}{\partial x_{3}}=u,$$

  where $a \in \mathbb{R}$, to be solved for $u=u\left(x_{1}, x_{2}, x_{3}\right)$. State clearly what it means for a hypersurface

  $$S_{\phi}=\left\{\left(x_{1}, x_{2}, x_{3}\right): \phi\left(x_{1}, x_{2}, x_{3}\right)=0\right\},$$

  defined by a $C^{1}$ function $\phi$, to be non-characteristic for $(*)$. Does the non-characteristic condition hold when $\phi\left(x_{1}, x_{2}, x_{3}\right)=x_{3}$ ?

  Solve $(*)$ for $a>0$ with initial condition $u\left(x_{1}, x_{2}, 0\right)=f\left(x_{1}, x_{2}\right)$ where $f \in C^{1}\left(\mathbb{R}^{2}\right)$. For the case $f\left(x_{1}, x_{2}\right)=x_{1}^{2}+x_{2}^{2}$ discuss the limiting behaviour as $a \rightarrow 0_{+}$.

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• 2.II.30C

  Partial Differential Equations | Part II, 2005

  Define a fundamental solution of a linear partial differential operator $P$. Prove that the function

  $$G(x)=\frac{1}{2} e^{-|x|}$$

  defines a distribution which is a fundamental solution of the operator $P$ given by

  $$P u=-\frac{d^{2} u}{d x^{2}}+u .$$

  Hence find a solution $u_{0}$ to the equation

  $$-\frac{d^{2} u_{0}}{d x^{2}}+u_{0}=V(x),$$

  where $V(x)=0$ for $|x|>1$ and $V(x)=1$ for $|x| \leqslant 1$.

  Consider the functional

  $$I[u]=\int_{\mathbb{R}}\left\{\frac{1}{2}\left[\left(\frac{d u}{d x}\right)^{2}+u^{2}\right]-V u\right\} d x .$$

  Show that $I\left[u_{0}+\phi\right]>I\left[u_{0}\right]$ for all Schwartz functions $\phi$ that are not identically zero.

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• 3.II.29C

  Partial Differential Equations | Part II, 2005

  Write down a formula for the solution $u=u(t, x)$ of the $n$-dimensional heat equation

  $$w_{t}(t, x)-\Delta w=0, \quad w(0, x)=g(x),$$

  for $g: \mathbb{R}^{n} \rightarrow \mathbb{C}$ a given Schwartz function; here $w_{t}=\partial_{t} w$ and $\Delta$ is taken in the variables $x \in \mathbb{R}^{n}$. Show that

  $$w(t, x) \leqslant \frac{\int|g(x)| d x}{(4 \pi t)^{n / 2}}$$

  Consider the equation

  $$u_{t}-\Delta u=e^{i t} f(x),$$

  where $f: \mathbb{R}^{n} \rightarrow \mathbb{C}$ is a given Schwartz function. Show that $(*)$ has a solution of the form

  $$u(t, x)=e^{i t} v(x),$$

  where $v$ is a Schwartz function.

  Prove that the solution $u(t, x)$ of the initial value problem for $(*)$ with initial data $u(0, x)=g(x)$ satisfies

  $$\lim _{t \rightarrow+\infty}\left|u(t, x)-e^{i t} v(x)\right|=0 .$$

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• 4.II.30C

  Partial Differential Equations | Part II, 2005

  Write down the solution of the three-dimensional wave equation

  $$u_{t t}-\Delta u=0, \quad u(0, x)=0, \quad u_{t}(0, x)=g(x),$$

  for a Schwartz function $g$. Here $\Delta$ is taken in the variables $x \in \mathbb{R}^{3}$ and $u_{t}=\partial u / \partial t$ etc. State the "strong" form of Huygens principle for this solution. Using the method of descent, obtain the solution of the corresponding problem in two dimensions. State the "weak" form of Huygens principle for this solution.

  Let $u \in C^{2}\left([0, T] \times \mathbb{R}^{3}\right)$ be a solution of

  $$u_{t t}-\Delta u+|x|^{2} u=0, \quad u(0, x)=0, \quad u_{t}(0, x)=0$$

  Show that

  $$\partial_{t} e+\nabla \cdot \mathbf{p}=0$$

  where

  $$e=\frac{1}{2}\left(u_{t}^{2}+|\nabla u|^{2}+|x|^{2} u^{2}\right), \quad \text { and } \quad \mathbf{p}=-u_{t} \nabla u .$$

  Hence deduce, by integration of $(* *)$ over the region

  $$K=\left\{(t, x): 0 \leqslant t \leqslant t_{0}-a \leqslant t_{0},\left|x-x_{0}\right| \leqslant t_{0}-t\right\}$$

  or otherwise, that $(*)$ satisfies the weak Huygens principle.

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2004

• B1.18

  Partial Differential Equations | Part II, 2004

  (a) State and prove the Mean Value Theorem for harmonic functions.

  (b) Let $u \geqslant 0$ be a harmonic function on an open set $\Omega \subset \mathbb{R}^{n}$. Let $B(x, a)=\{y \in$ $\left.\mathbb{R}^{n}:|x-y|<a\right\}$. For any $x \in \Omega$ and for any $r>0$ such that $B(x, 4 r) \subset \Omega$, show that

  $$\sup _{\{y \in B(x, r)\}} u(y) \leqslant 3^{n} \inf _{\{y \in B(x, r)\}} u(y) .$$

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• B2.18

  Partial Differential Equations | Part II, 2004

  (a) State and prove the Duhamel principle for the wave equation.

  (b) Let $u \in C^{2}\left([0, T] \times \mathbb{R}^{n}\right)$ be a solution of

  $$u_{t t}+u_{t}-\Delta u+u=0$$

  where $\Delta$ is taken in the variables $x \in \mathbb{R}^{n}$ and $u_{t}=\partial_{t} u$ etc.

  Using an 'energy method', or otherwise, show that, if $u=u_{t}=0$ on the set $\left\{t=0,\left|x-x_{0}\right| \leqslant t_{0}\right\}$ for some $\left(t_{0}, x_{0}\right) \in[0, T] \times \mathbb{R}^{n}$, then $u$ vanishes on the region $K(t, x)=\left\{(t, x): 0 \leqslant t \leqslant t_{0},\left|x-x_{0}\right| \leqslant t_{0}-t\right\}$. Hence deduce uniqueness for the Cauchy problem for the above PDE with Schwartz initial data.

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• B3.18

  Partial Differential Equations | Part II, 2004

  (i) Find $w:[0, \infty) \times \mathbb{R} \longrightarrow \mathbb{R}$ such that $w(t, \cdot)$ is a Schwartz function of $\xi$ for each $t$ and solves

  $$\begin{gathered} w_{t}(t, \xi)+\left(1+\xi^{2}\right) w(t, \xi)=g(\xi), \\ w(0, \xi)=w_{0}(\xi), \end{gathered}$$

  where $g$ and $w_{0}$ are given Schwartz functions and $w_{t}$ denotes $\partial_{t} w$. If $\mathcal{F}$ represents the Fourier transform operator in the $\xi$ variables only and $\mathcal{F}^{-1}$ represents its inverse, show that the solution $w$ satisfies

  $$\partial_{t}\left(\mathcal{F}^{-1}\right) w(t, x)=\mathcal{F}^{-1}\left(\partial_{t} w\right)(t, x)$$

  and calculate $\lim _{t \rightarrow \infty} w(t, \cdot)$ in Schwartz space.

  (ii) Using the results of Part (i), or otherwise, show that there exists a solution of the initial value problem

  $$\begin{gathered} u_{t}(t, x)-u_{x x}(t, x)+u(t, x)=f(x) \\ u(0, x)=u_{0} \end{gathered}$$

  with $f$ and $u_{0}$ given Schwartz functions, such that

  $$\|u(t, \cdot)-\phi\|_{L^{\infty}(\mathbb{R})} \longrightarrow 0$$

  as $t \rightarrow \infty$ in Schwartz space, where $\phi$ is the solution of

  $$-\phi^{\prime \prime}+\phi=f .$$

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• B4.18

  Partial Differential Equations | Part II, 2004

  (a) State a theorem of local existence, uniqueness and $C^{1}$ dependence on the initial data for a solution for an ordinary differential equation. Assuming existence, prove that the solution depends continuously on the initial data.

  (b) State a theorem of local existence of a solution for a general quasilinear firstorder partial differential equation with data on a smooth non-characteristic hypersurface. Prove this theorem in the linear case assuming the validity of the theorem in part (a); explain in your proof the importance of the non-characteristic condition.

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2003

• B1.18

  Partial Differential Equations | Part II, 2003

  (a) Define characteristic hypersurfaces and state a local existence and uniqueness theorem for a quasilinear partial differential equation with data on a non-characteristic hypersurface.

  (b) Consider the initial value problem

  $$3 u_{x}+u_{y}=-y u, \quad u(x, 0)=f(x),$$

  for a function $u: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $C^{1}$ initial data $f$ given for $y=0$. Obtain a formula for the solution by the method of characteristics and deduce that a $C^{1}$ solution exists for all $(x, y) \in \mathbb{R}^{2}$.

  Derive the following (well-posedness) property for solutions $u(x, y)$ and $v(x, y)$ corresponding to data $u(x, 0)=f(x)$ and $v(x, 0)=g(x)$ respectively:

  $$\sup _{x}|u(x, y)-v(x, y)| \leqslant \sup _{x}|f(x)-g(x)| \quad \text { for all } y .$$

  (c) Consider the initial value problem

  $$3 u_{x}+u_{y}=u^{2}, \quad u(x, 0)=f(x),$$

  for a function $u: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $C^{1}$ initial data $f$ given for $y=0$. Obtain a formula for the solution by the method of characteristics and hence show that if $f(x)<0$ for all $x$, then the solution exists for all $y>0$. Show also that if there exists $x_{0}$ with $f\left(x_{0}\right)>0$, then the solution does not exist for all $y>0$.

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• B2.17

  Partial Differential Equations | Part II, 2003

  (a) If $f$ is a radial function on $\mathbb{R}^{n}$ (i.e. $f(x)=\phi(r)$ with $r=|x|$ for $x \in \mathbb{R}^{n}$ ), and $n>2$, then show that $f$ is harmonic on $\mathbb{R}^{n}-\{0\}$ if and only if

  $$\phi(r)=a+b r^{2-n}$$

  for $a, b \in \mathbb{R}$.

  (b) State the mean value theorem for harmonic functions and prove it for $n>2$.

  (c) Generalise the statement and the proof of the mean value theorem to the case of a subharmonic function, i.e. a $C^{2}$ function such that $\Delta u \leqslant 0$.

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• B3.18

  Partial Differential Equations | Part II, 2003

  Consider the initial value problem

  $$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=0$$

  to be solved for $u: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$, subject to the initial conditions

  $$u(0, x)=f(x), \quad \frac{\partial u}{\partial t}(0, x)=0$$

  for $f$ in the Schwarz space $\mathcal{S}\left(\mathbb{R}^{n}\right)$. Use the Fourier transform in $x$ to obtain a representation for the solution in the form

  $$u(t, x)=\int e^{i x \cdot \xi} A(t, \xi) \widehat{f}(\xi) d^{n} \xi$$

  where $A$ should be determined explicitly. Explain carefully why your formula gives a smooth solution to (1) and why it satisfies the initial conditions (2), referring to the required properties of the Fourier transform as necessary.

  Next consider the case $n=1$. Find a tempered distribution $T$ (depending on $t, x$ ) such that (3) can be written

  $$u=<T, \widehat{f}>$$

  and (using the definition of Fourier transform of tempered distributions) show that the formula reduces to

  $$u(t, x)=\frac{1}{2}[f(x-t)+f(x+t)]$$

  State and prove the Duhamel principle relating to the solution of the $n$-dimensional inhomogeneous wave equation

  $$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=h$$

  to be solved for $u: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$, subject to the initial conditions

  $$u(0, x)=0, \quad \frac{\partial u}{\partial t}(0, x)=0$$

  for $h$ a $C^{\infty}$ function. State clearly assumptions used on the solvability of the homogeneous problem.

  [Hint: it may be useful to consider the Fourier transform of the tempered distribution defined by the function $\xi \mapsto e^{i \xi \cdot a}$.]

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• B4.18

  Partial Differential Equations | Part II, 2003

  Discuss the basic properties of the Fourier transform and how it is used in the study of partial differential equations.

  The essay should include: definition and basic properties, inversion theorem, applications to establishing well-posedness of evolution partial differential equations with constant coefficients.

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2002

• B1.18

  Partial Differential Equations | Part II, 2002

  (a) Solve the equation, for a function $u(x, y)$,

  $$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$$

  together with the boundary condition on the $x$-axis:

  $$u(x, 0)=x$$

  Find for which real numbers $a$ it is possible to solve $(*)$ with the following boundary condition specified on the line $y=a x$ :

  $$u(x, a x)=x$$

  Explain your answer in terms of the notion of characteristic hypersurface, which should be defined.

  (b) Solve the equation

  $$\frac{\partial u}{\partial x}+(1+u) \frac{\partial u}{\partial y}=0$$

  with the boundary condition on the $x$-axis

  $$u(x, 0)=x$$

  in the domain $\mathcal{D}=\left\{(x, y): 0<y<(x+1)^{2} / 4,-1<x<\infty\right\}$. Sketch the characteristics.

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• B2.17

  Partial Differential Equations | Part II, 2002

  (a) Define the convolution $f * g$ of two functions. Write down a formula for a solution $u:[0, \infty) \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ to the initial value problem

  $$\frac{\partial u}{\partial t}-\Delta u=0$$

  together with the boundary condition

  $$u(0, x)=f(x)$$

  for $f$ a bounded continuous function on $\mathbb{R}^{n}$. Comment briefly on the uniqueness of the solution.

  (b) State and prove the Duhamel principle giving the solution (for $t>0$ ) to the equation

  $$\frac{\partial u}{\partial t}-\Delta u=g$$

  together with the boundary condition

  $$u(0, x)=f(x)$$

  in terms of your answer to (a).

  (c) Show that if $v:[0, \infty) \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ is the solution to

  $$\frac{\partial v}{\partial t}-\Delta v=G$$

  together with the boundary condition

  $$v(0, x)=f(x)$$

  with $G(t, x) \leq g(t, x)$ for all $(t, x)$ then $v(t, x) \leq u(t, x)$ for all $(t, x) \in(0, \infty) \times \mathbb{R}^{n}$.

  Finally show that if in addition there exists a point $\left(t_{0}, x_{0}\right)$ at which there is strict inequality in the assumption i.e.

  $$G\left(t_{0}, x_{0}\right)<g\left(t_{0}, x_{0}\right)$$

  then in fact

  $$v(t, x)<u(t, x)$$

  whenever $t>t_{0}$.

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• B3.18

  Partial Differential Equations | Part II, 2002

  Define the Schwartz space $\mathcal{S}\left(\mathbb{R}^{n}\right)$ and the space of tempered distributions $\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$. State the Fourier inversion theorem for the Fourier transform of a Schwartz function.

  Consider the initial value problem:

  $$\begin{gathered} \frac{\partial^{2} u}{\partial t^{2}}-\Delta u+u=0, x \in \mathbb{R}^{n}, 0<t<\infty \\ u(0, x)=f(x), \quad \frac{\partial u}{\partial t}(0, x)=0 \end{gathered}$$

  for $f$ in the Schwartz space $\mathcal{S}\left(\mathbb{R}^{n}\right)$.

  Show that the solution can be written as

  $$u(t, x)=(2 \pi)^{-n / 2} \int_{\mathbb{R}^{n}} e^{i x \cdot \xi} \hat{u}(t, \xi) d \xi,$$

  where

  $$\hat{u}(t, \xi)=\cos \left(t \sqrt{1+|\xi|^{2}}\right) \hat{f}(\xi)$$

  and

  $$\hat{f}(\xi)=(2 \pi)^{-n / 2} \int_{\mathbb{R}^{n}} e^{-i x \cdot \xi} f(x) d x .$$

  State the Plancherel-Parseval theorem and hence deduce that

  $$\int_{\mathbb{R}^{n}}|u(t, x)|^{2} d x \leq \int_{\mathbb{R}^{n}}|f(x)|^{2} d x .$$

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• B4.18

  Partial Differential Equations | Part II, 2002

  Discuss the notion of fundamental solution for a linear partial differential equation with constant coefficients.

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2001

• B1.18

  Partial Differential Equations | Part II, 2001

  (a) Solve the equation

  $$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=u^{2}$$

  together with the boundary condition on the $x$-axis:

  $$u(x, 0)=f(x)$$

  where $f$ is a smooth function. You should discuss the domain on which the solution is smooth. For which functions $f$ can the solution be extended to give a smooth solution on the upper half plane $\{y>0\}$ ?

  (b) Solve the equation

  $$x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=0$$

  together with the boundary condition on the unit circle:

  $$u(x, y)=x \quad \text { when } \quad x^{2}+y^{2}=1$$

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• B2.17

  Partial Differential Equations | Part II, 2001

  Define the Schwartz space $\mathcal{S}(\mathbb{R})$ and the corresponding space of tempered distributions $\mathcal{S}^{\prime}(\mathbb{R})$.

  Use the Fourier transform to give an integral formula for the solution of the equation

  $$-\frac{d^{2} u}{d x^{2}}+\frac{d u}{d x}+u=f$$

  for $f \in \mathcal{S}(\mathbb{R})$. Prove that your solution lies in $\mathcal{S}(\mathbb{R})$. Is your formula the unique solution to $(*)$ in the Schwartz space?

  Deduce from this formula an integral expression for the fundamental solution of the operator $P=-\frac{d^{2}}{d x^{2}}+\frac{d}{d x}+1$.

  Let $K$ be the function:

  $$K(x)= \begin{cases}\frac{1}{\sqrt{5}} e^{-(\sqrt{5}-1) x / 2} & \text { for } x \geqslant 0 \\ \frac{1}{\sqrt{5}} e^{(\sqrt{5}+1) x / 2} & \text { for } x \leqslant 0\end{cases}$$

  Using the definition of distributional derivatives verify that this function is a fundamental solution for $P$.

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• B3.18

  Partial Differential Equations | Part II, 2001

  Write down a formula for the solution $u=u(t, x)$, for $t>0$ and $x \in \mathbb{R}^{n}$, of the initial value problem for the heat equation:

  $$\frac{\partial u}{\partial t}-\Delta u=0 \quad u(0, x)=f(x)$$

  for $f$ a bounded continuous function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. State (without proof) a theorem which ensures that this formula is the unique solution in some class of functions (which should be explicitly described).

  By writing $u=e^{t} v$, or otherwise, solve the initial value problem

  $$\tag{†} \frac{\partial v}{\partial t}+v-\Delta v=0, \quad v(0, x)=g(x)$$

  for $g$ a bounded continuous function $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and give a class of functions in which your solution is the unique one.

  Hence, or otherwise, prove that for all $t>0$ :

  $$\sup _{x \in \mathbb{R}^{n}} v(t, x) \leqslant \sup _{x \in \mathbb{R}^{n}} g(x)$$

  and deduce that the solutions $v_{1}(t, x)$ and $v_{2}(t, x)$ of $(†)$ corresponding to initial values $g_{1}(x)$ and $g_{2}(x)$ satisfy, for $t>0$,

  $$\sup _{x \in \mathbb{R}^{n}}\left|v_{1}(t, x)-v_{2}(t, x)\right| \leqslant \sup _{x \in \mathbb{R}^{n}}\left|g_{1}(x)-g_{2}(x)\right| .$$

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• B4.18

  Partial Differential Equations | Part II, 2001

  Write an essay on one of the following two topics:

  (a) The notion of well-posedness for initial and boundary value problems for differential equations. Your answer should include a definition and give examples and state precise theorems for some specific problems.

  (b) The concepts of distribution and tempered distribution and their use in the study of partial differential equations.

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