Strong solutions of the double phase parabolic equations with variable growth

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 \[ (\epsilon^2+|\nabla u|^2)^{\frac{p(z)-2}{2}}\nabla u + a(z) (\epsilon^2+|\nabla u|^2)^{\frac{q(z)-2}{2}}\nabla u, \qquad \epsilon>0. \]