Ir al contenido principal

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 $\Omega\times(0,T]$.