It is well known that a weak solution to linear parabolic equation $u_t - div(A(x)\nabla u) = 0$ with homogeneous Neumann boundary condition, is Holder continuous with $A(x)$ belongs only to $L^\infty(\Omega)$ and satisfies the uniform elliptic condition.

My question is that: can we say the same for a weak subsolution, i.e. $$\int_0^T\int_{\Omega}u_t\eta + A(x)\nabla u \nabla \eta \leq 0$$ for all non-negative test functions $\eta\geq 0$? Assume additionally that we have the subsolution is non-negative $u\geq 0$ and uniformly bounded $\|u\|_{L^\infty(\Omega\times (0,T))} \leq M$.

If it can not be true, how about the Laplacian case, i.e. when $A(x) = Id$.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.