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.