Problem set 6, PDE

Problem 1

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

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

as t\to\infty.

Problem 2

For any 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

\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 \partial_t^2 - \partial_x^2 on \R^2 commutes with the Lorentz transformations (hyperbolic rotations)

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

for \theta\in\R.

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