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Problem set 13, PDE

Problem 1

Let $latex \Omega\subset\R^n$ be a bounded domain and $latex u_0\in C(\bar\Omega)$. Suppose that $latex u\in C^{2,1}(\Omega\times(0,\infty))\cap C(\bar\Omega\times[0,\infty))$ is a solution of

$latex \begin{cases} \partial_t u - \Delta u=0 & \text{in }\Omega\times(0,\infty)\\ u(\cdot,0) = u_0 & \text{on } \Omega \\ u=0 & \text{on } \partial\Omega\times(0,\infty).\end{cases}$

Then there exist constants $latex \mu, C>0$, depending only on $latex n,\Omega$, such that

$latex \displaystyle \sup_\Omega |u(\cdot,t)| \le C e^{-\mu t} \sup_\Omega |u_0|$

for any $latex t>0$.

Problem 2

Let $latex \Omega\subset\R^n$ be a bounded domain, $latex c\in C(\bar\Omega\times[0,T])$ with $latex c\ge -c_0$ for a constant $latex c_0\ge 0$, and $latex u_0\in C(\Omega)$ nonnegative. Suppose $latex u\in C^{2,1}(\Omega\times(0,T])\cap C(\bar\Omega\times[0,T])$ is a solution of

$latex \begin{cases} \partial_t u - \Delta u + cu = -u^2 & \text{in }\Omega\times(0,T]\\ u(\cdot,0) = u_0 & \text{on } \Omega \\ u=0 & \text{on } \partial\Omega\times(0,T).\end{cases}$

Then $latex 0 \le u \le e^{c_0 T} \sup_\Omega u_0$ in $latex \Omega\times(0,T]$.