Let $\Omega={(x,y)\in\mathbb{R}^2: x^2+y^2<4}$ and $\Gamma={(x,y)\in\mathbb{R}^2:x^2+y^2=4}$. We consider the problem of finding a solution $u:\Omega\times(0,T)\to \mathbb{R}$ of the system \begin{align*} u_t-\operatorname{div}(A_\alpha\nabla u)&=y\,,(x,y)\in \Omega\times (0,T),\\ u(x,y,0)&=0\,,(x,y)\in \Omega,\\ u&=0\,,(x,y)\in \Gamma\times (0,T), \end{align*} where $$A_\alpha(x,y)=\begin{pmatrix} 1&0 \\ 0&\alpha e^{x^2+y^2} \end{pmatrix}.$$ Im are asked to calculate for which values of $\alpha$ the problem is parabolic, formulate the associated variational problem and apply the Faedo-Galerkin method to prove the existence and uniqueness of the problem.\
I managed to prove that the problem is parabolic for $\alpha>0$ and that the variational problem is $(u_t,v)+B(u,v)=Tv$ for $B(u,v)=\int (A\nabla u)\cdot \nabla v$ and $Tv=\int yv$. But im completely lost with the Faedo-Galerkin part... Any help will be very appreciated
The Faedo-Galerkin method asserts that to demonstrate the existence of a solution, it is necessary to prove that the solution is the limit of an orthogonal and complete system in $H^1(\Omega)$. This is expressed as:
\begin{align} u(t,\boldsymbol{x}) = \sum_{i=1}^{\infty} c_i(t) u_i(\boldsymbol{x}) \end{align} The most suitable approach is to select $\{u_j\}_j$ as the eigenfunctions of the operator induced by the bilinear form $B$.
To achieve this, we must first define for any positive integer $n$ \begin{align*} u^{(n)} = \sum_{i=1}^{n} c_i u_i \end{align*} and then show that the problem $ (u^{(n)}_t, v) + B(u^{(n)}, v) = Tv $ can be solved. This is feasible due to the ellipticity of $B$; indeed, it results in a first order linear ODE system that grows as $n$ increases.
We then demonstrate that the sequences $ \{u^{(n)}\}_n$ and $ \{\dot{u}^{(n)}\}_n $ are bounded, implying the existence of subsequences that converge weakly. Using this weak convergence, we prove that the limit $ \lim_n u^{(n)} $ satisfies the equation $(u^n_t, v) + B(u^n, v) = Tv $.
More technical details can be found at the following article.