Let $u$ denote to the solution of the heat equation
$\begin{cases} u_t(x,t)-\Delta u(x,t) & = & 0 & t>0 \\ u(x,0) & = & g(x) \end{cases}$
where $x\in\mathbb{R}^n$.
I want to show that
- if $||g||_\infty<\infty$ then $u$ tends to some constant as $t\to\infty$
- if $\text{supp}(g) \Subset \mathbb{R}$ (that is compact) then this constant is $0$.
I started from $u(x,t) = (g*K_t)(x) \ \ \ \ \ \ \ \ \ (\diamond)$ where $K_t$ is the heat kernel and tried to prove 1. by showing $||u'(x,t)||_{\infty}\to 0$ as $t\to\infty$. Unfortunately this might be the wrong way since formula $(\diamond)$ solves the heat equation (with initial data) only if $g \in L^p$ (what is not clear as we only claim $g$ to be bounded).
Who can help?