(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 $t>0$.