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

TL;DR

This paper establishes global L^r-estimates and regularizing effects for solutions to p(t,x)-Laplacian systems, demonstrating uniform regularity and maximum modulus properties under certain conditions.

Contribution

It provides new global L^r-estimates and regularity results for p(t,x)-Laplacian systems, extending understanding of solution behavior with variable exponents.

Findings

Global L^{r_0} regularity for solutions with initial data in L^{r_0}

Maximum modulus theorem for the case r_0 = 

L^{r_0}-L^r estimates under p- > 2n/(n+r_0)

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 give a global L^{r_0}({\Omega})-regularity result uniformly in t>0 that, in the particular case r_0 =\infty, implies a maximum modulus theorem. Under the assumption p- = \inf p(t, x) > 2n/(n+r_0), we also state L^{r_0}- L^r estimates for the solution, for r \geq r_0. Complete proofs of the results presented here are given in the paper [F. Crispo, P. Maremonti, M. Ruzicka, Global L^r-estimates and regularizing effect for solutions to the p(t, x) -Laplacian systems, accepted for publication on Advances in Differential Equations, 2017].