Very weak solutions of subquadratic parabolic systems with non-standard $p(x,t)$-growth

TL;DR

This paper proves higher integrability of very weak solutions for parabolic systems with non-standard subquadratic $p(x,t)$-growth, extending previous results to the case where the growth is less than quadratic.

Contribution

It establishes higher integrability for very weak solutions of parabolic $p(x,t)$-Laplacian systems with subquadratic growth, extending prior results to this new regime.

Findings

Very weak solutions belong to natural energy spaces under certain conditions.

Higher integrability is achieved for solutions with subquadratic $p(x,t)$-growth.

Extension of previous results to the subquadratic case.

Abstract

The aim of this paper is to establish a higher integrability result for very weak solutions of certain parabolic systems whose model is the parabolic $p(x,t)$-Laplacian system. Under assumptions on the exponent function $p:\Omega_T=\Omega\times (0,T)\to\left(\frac{2n}{n+2},2\right]$, it is shown that any very weak solution $u:\Omega_T\rightarrow\mathbb{R}^N$ with $|Du|^{p(\cdot)(1-\varepsilon)}\in L^1(\Omega_T)$ belongs to the natural energy spaces, i.e. $|Du|^{p(\cdot)}\in L^1_{\operatorname{loc}}(\Omega_T)$, provided $\epsilon>0$ is small enough. This extends the main result of [V. B\"ogelein and Q. Li, Nonlinear Anal., 98 (2014), pp. 190-225] to the subquadratic case.