Problem set 13, PDE

Problem 1

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

\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 \mu, C>0, depending only on n,\Omega, such that

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

for any t>0.

Problem 2

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

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

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