Let $n\geq 2$ be an integer and $\Omega\in R^n$ be a bounded domain with boundary $\partial\Omega$ . Consider the following IBVP:
$u_t=\Delta u$, for $x\in \Omega$, $t>0$;
$u(x, 0)=f(x), x\in\Omega$ (intial condition )
$u= g(x,t), x\in\partial\Omega, t>0$ (boundary condition).
When $n=1$ we can easily homogenize the boundary condition and then use separation of variables or any other standard method to show the existence of the solution. When $n\geq 2$ and $g(x, t)=0$ (means zero boundary condition) we can use Galerkin's method to prove the existence of the solution. I try to understand how to prove the existence of above problem if $g(x,t)\neq 0$ and if $n\geq 2$.
Of course the above problem is just an example to explain my point. The equation could be any nonlinear partial differential differential equation. In my previous post I asked if there is a method to handle for such problems (problems with nonhomogeneous boundary condition in higher dimensions as given above). Any help is appreciated.