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Problem set 6, PDE

Problem 1

Let $latex u_0\in L^2(0,\pi)$ be the solution of the heat equation seen in class. Then, for all $latex i,j\in\N$,

$latex \displaystyle \sup_{x\in[0,\pi]}|\partial_x^i \partial_t^j u(x,t)|\to 0$

as $latex t\to\infty$.

Problem 2

For any $latex u_0\in L^2(0,\pi), f\in L^2((0,\pi)\times(0,\infty))$, find a formal expression of a solution of the problem

$latex \displaystyle \begin{cases} \partial_t u - \partial^2_x u = f & \text{in }(0,\pi)\times(0,\infty)\\ u(x,0) = u_0(x) & x\in (0,\pi)\\ u(0,t)=u(\pi,t)=0 & t\in (0,\infty). \end{cases}$

Problem 3

The wave operator $latex \partial_t^2 - \partial_x^2$ on $latex \R^2$ commutes with the Lorentz transformations (hyperbolic rotations)

$latex \displaystyle T_\theta = \begin{pmatrix}\cosh\theta & \sinh\theta\\ \sinh\theta & \cosh\theta\end{pmatrix},$

for $latex \theta\in\R$.