ap.analysis of pdes – Perturbation in the equation $u_t=epsilon Pu$, where $P$ is an elliptic partial differential operator


Let $Pu=sum_{ij} partial_j(a_{ij}(x) partial_{i} u)$ be an elliptic operator. Consider the equation
$$
(u=u_epsilon)\
partial_t u=epsilon Pu text{ in } mathbb R^+ times Omega,\
u(0,x)=u_0(x), u(t,x)=0text{ for }x in partialOmega
$$

where $Omega$ is an open set in $mathbb R^n$ (not necessarily bounded).

A weak solution to this equation exists and is unique in $L^2(0,T,H_0^1(Omega))$.

Let $u_epsilon$ be the solution to the equation corresponding to the given value of $epsilon$. I wish to show that when $epsilon to 0$, $u_epsilon$ converges in $L^2$ to a limit $u$.

By checking the formulation of a weak solution ($chi$ is a test function, for example, in $C^1$):
$$
int u_epsilon partial_t chi=-epsilonint (nabla u) A (nablachi),
$$

we could show that $u_epsilon$ is bounded in the $H^1$ norm for $epsilon<1/2$, and hence that
$$
int (partial_tu_{epsilon_1}-partial_tu_{epsilon_2}) chi to 0
$$

as $epsilon_1,epsilon_2 to 0$, for every test function $chi$. Hence $u_epsilon$ converges to a distributional limit.

Now, is it ture that $u_epsilon to u$ in $L^infty(0,T,W_{1,infty}(Omega))$? This is true in the case of $Pu=Delta u$ by direct computation, after writing out the solution explicitly as an integral (heat equation). But in the general case, how could I prove that $partial_j u_epsilon$ converges a.e. to a limit $partial_j u$?