# 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 $$partial_j u$$?