Math 6410 • ODE

Heat Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary \(\partial \Omega \). Let \(f\in C^2(\overline {\Omega })\), \(g\in C^\infty (\partial \Omega )\), and assume \(f=g\) on \(\partial \Omega \). Let \(u(x,t)\) solve \[ \begin {aligned} u_t &= \Delta u,\quad x\in \Omega ,\quad t>0,\\ u(x,0) &= f(x),\quad x\in \overline {\Omega },\\ u(x,t) &= g(x),\quad x\in \partial \Omega ,\quad t>0. \end {aligned} \] Prove that \[ \lim _{t\to \infty }\int _\Omega \lvert u(x,t)-v(x)\rvert ^2\,dx=0 \] where \(v\) solves \[ \begin {aligned} \Delta v &= 0,\quad \text {in }\Omega ,\\ v &= g,\quad \text {on }\partial \Omega . \end {aligned} \]