Global gradient estimates for solutions of parabolic equations with nonstandard growth

Abstract

We study how the smoothness of the initial datum and the free term affect the global regularity properties of solutions to the Dirichlet problem for the class of parabolic equations of $p(x,t)$-Laplace type %with nonlinear sources depending on the solution and its gradient: \[ u_t-\Delta_{p(\cdot)}u=f(z)+F(z,u,\nabla u),\quad z=(x,t)\in Q_T=\Omega\times (0,T), \] with the nonlinear source $F(z,u,\nabla u)=a(z)|u|^{q(z)-2}u+|\nabla u|^{s(z)-2}(\vec c,\nabla u)$. It is proven the existence of a solution such that if $|\nabla u(x,0)|\in L^r(\Omega)$ for some $r\geq \max\{2,\max p(z)\}$, then the gradient preserves the initial order of integrability in time, gains global higher integrability, and the solution acquires the second-order regularity in the following sense: \[ \text{$|\nabla u(x,t)|\in L^r(\Omega)$ for a.e. $t \in (0,T)$}, \qquad \text{$|\nabla u|^{p(z)+\rho+r-2} \in L^1(Q_T)$ for any $\rho \in \left(0, \frac{4}{N+2}\right)$}, \] and \[ |\nabla u|^{\frac{p(z)+r}{2}-2}\nabla u\in L^2(0,T;W^{1,2}(\Omega))^N. \] The exponent $r$ is arbitrary and independent of $p(z)$ if $f\in L^{N+2}(Q_T)$, while for $f\in L^\sigma(Q_T)$ with $\sigma \in (2,N+2)$ the exponent $r$ belongs to a bounded interval whose endpoints are defined by $\max p(z)$, $\min p(z)$, $N$, and $\sigma$. An integration by parts formula is also proven, which is of independent interest.