Given the heat equation
$u_t=au_{xx}+cu$
$u(x,0)=f(x)\geq0$
$u(0,t)=u(1,t)=0$
find apriori bound on c that doesn't grow for every $f$ without solving.
my attempt:
multiplying by $u$ and integrating on x from 0 to 1
$\int_{0}^{1}uu_tdx=\int_{0}^{1}auu_{xx}+cu^2dx\iff \frac{\partial}{\partial t}\frac{1}{2}||u||^2=c||u||^2+a[uu_x|_{0}^{1}-\int_{0}^{1}(u_x)^2]\iff \frac{\partial}{\partial t}\frac{1}{2}||u||^2=c||u||^2+a[uu_x|_{0}^{1}-||u_x||^2]\iff\frac{\partial}{\partial t}\frac{1}{2}||u||^2=c||u||^2+a[-||u_x||^2]$ $\iff \frac{\partial}{\partial t}\frac{1}{2}||u||^2\leq ||u||^2(c-a\pi^2)$
multiply by $e^{2(c-a\pi^2)}$ we can integrate and get:
$ =(e^{2(c-a\pi^2)}||u||^2)_t\to e^{2(c-a\pi^2)}||u(*,t)||^2-||u(*,0)||^2\leq0$ so we need $c-a\pi^2\geq0$ for the answer not to grow fast for any f.
and $uu_x|_{0}^{1}=0$ from the boundary conditions
not sure about the boundary condition use and the inequality