Reference for Comparison Principle

Question

I know that there is the following theorem: $$ \mathcal{L}u = u_t +Lu\\ Lu = -a^{i,j}(x,t,u)u_{x_i,x_j} + b^i(x,t,u)u_{x_i} + c(x,t,u)u $$ Suppose $\mathcal{L}$ uniformly parabolic and $a^{i,j}, b^i, c$ smooth. Let $u,v \in C^{2,1}(Q_T)$ be such that: $$ \mathcal{L}u \leq 0 \text{ in } Q_T\\ \mathcal{L}v \geq 0 \text{ in } Q_T\\ u\leq v \text{ on } \Gamma_T $$ with $\Omega \in \mathbb{R}^N$ open, bounded and connected, $Q_T = \Omega \times(0,T]$ the usual parabolic cylinder and $\Gamma_T$ its parabolic boundary. Then: $$ u\leq v \text{ in } Q_T $$ It should be called comparison principle. Do you know where can I find it? I was also wondering if there is a version of it that does not require the sub/super-solutions to be $C^{2,1}(Q_T)$ (maybe one that holds for weak sub/super-solutions?) Thank you very much in advance.

Answer 1

You could have a look at Theorem 13.5 in "Abstract Evolution Equations, Periodic Problems and Applications" by Daniel Daners and Pablo Koch Medina.

They deal with weak formulations, so $u \in W^{2,1,p}(\Omega\times (0,T))$ is necessary. $W^{2,1,p}(\Omega)$ is the space consisting of those functions which posess spatial weak derivatives up to second order and temporal weak derivatives of first order (all in $L^p$).