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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 $A(x) = Id$.

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