Higher integrability for parabolic equations of $p(x,t)$-Laplacian type

Abstract

Let $\Omega\subset\mathbb{R}^{n}$, $n\geq2$, be a bounded domain withboundary $\partial\Omega$, and $Q=\Omega\times(0,T]$ be a cylinder of height $Ts < \infty$. We study local weak solutions of the parabolic equation \[ Lu\equiv\frac{\partial u}{\partial t}-div\left( \left| \nabla u\right| ^{p(z)-2}\nabla u\right)=0,\quad z=(x,t)\in\Omega\times(0,T), \] with variable exponent of nonlinearity $p$. We assume that $p(z)\in C(\Omega)$ and is such that \[ \frac{2n}{n+2}s < \alpha\leq p(z)\leq\beta < \infty,\quad z\in Q, \] \[ \left| p(z_{1})-p(z_{2})\right| \leq\omega(|z_{1}-z_{2}|)\quad \forall \,(z_{1},\,z_{2})\in\overline{Q}, \] \[ \overline{\lim_{\tau\rightarrow0}}\quad \omega(\tau)\ln\frac{1}{\tau}s <\infty. \] We prove that the weak solution is bounded and establish Meyer's type estimates: there exists a positive constant $\varepsilon>0$ such that for every subdomain $Q^{\prime}$, $\overline{Q^{\prime}}\subset Q$, \[ \int_{Q^{\prime}}\left| \nabla u\right| ^{p(z)(1+ \varepsilon )}dz s <\infty. \]