Strong solutions of the double phase parabolic equations with variable growth (2010.08306v2)

Abstract: This paper addresses the questions of existence and uniqueness of strong solutions to the homogeneous Dirichlet problem for the double phase equation with operators of variable growth: [ u_t - div \left(|\nabla u|^{p(z)-2} \nabla u+ a(z) |\nabla u|^{q(z)-2} \nabla u \right) = F(z,u) \quad \text{in $Q_T=\Omega \times (0,T)$} ] where $\Omega \subset \mathbb{R}^N$, $N \geq 2$, is a bounded domain with the boundary $\partial\Omega\in C^2$, $z=(x,t)\in Q_T$, $a:\bar Q_T \mapsto \mathbb{R}$ is a given nonnegative coefficient, and the nonlinear source term has the form [ F(z,v)=f_0(z)+b(z)|v|^{\sigma(z)-2}v. ] The variable exponents $p$, $q$, $\sigma$ are given functions defined on $\bar{Q}T$, $p$, $q$ are Lipschitz-continuous and [ \dfrac{2N}{N+2}<p^-\leq p(z) \leq q(z) < p(z) + {\frac{r}{2}} \ \ \text{with $0<r<r^\ast=\frac{4p^-}{2N + p^-(N+2)}$,\quad $p^-=\min{\bar{Q}_T}p(z)$}. ] We find conditions on the functions $f_0$, $a$, $b$, $\sigma$ and $ u_0$ sufficient for the existence of a unique strong solution with the following global regularity and integrability properties: [ \begin{split} u_t \in L^{2}(Q_T),\quad & \text{$|\nabla u|^{s(z)} \in L^{\infty}(0,T;L^1(\Omega))$ with $ s(z)=\max{2,p(z)}$}, & |\nabla u|^{p(z)+\delta}\in L^1(Q_T)\quad \text{for every $0<\delta< r^*$}. {split} ] The same results are established for the equation with the regularized flux [ (\epsilon^2+|\nabla u|^2)^{\frac{p(z)-2}{2}}\nabla u + a(z) (\epsilon^2+|\nabla u|^2)^{\frac{q(z)-2}{2}}\nabla u, \qquad \epsilon>0. ]