Global $L^r$-estimates and regularizing effect for solutions to the $p(t,x)$-Laplacian systems

Abstract

We consider the initial boundary value problem for the $p(t,x)$-Laplacian system in a bounded domain $\Omega$. If the initial data belongs to $L^{r_0}$, $r_0\geq 2$, we prove a global $L^{r_0}(\Omega)$-regularity result uniformly in $t>0$ that, in the particular case ${r_0}=\infty$, gives a maximum modulus theorem. Under the assumption $p_-=\inf p(t,x)>\frac{2n} {n+r_0}$, we also study $L^{r_0}-L^{r}$ estimates for the solution, for $r\geq r_0$.