Optimal global second-order regularity and improved integrability for parabolic equations with variable growth

TL;DR

This paper establishes optimal regularity and integrability results for solutions to variable exponent parabolic equations with nonlinear sources, advancing understanding of their mathematical properties under certain conditions.

Contribution

It proves the existence of solutions with optimal regularity and improved integrability for variable exponent parabolic equations with nonlinear sources.

Findings

Solutions exist with bounded gradient integrability.

Gradient belongs to Sobolev space W^{1,2}.

Global regularity properties are established.

Abstract

We consider the homogeneous Dirichlet problem for the parabolic equation \[ u_t- \operatorname{div} \left(|\nabla u|^{p(x,t)-2} \nabla u\right)= f(x,t) + F(x,t, u, \nabla u) \] in the cylinder $Q_T:=\Omega\times (0,T)$, where $\Omega\subset \mathbb{R}^N$, $N\geq 2$, is a $C^{2}$-smooth or convex bounded domain. It is assumed that $p\in C^{0,1}(\overline{Q}_T)$ is a given function, and that the nonlinear source $F(x,t,s, \xi)$ has a proper power growth with respect to $s$ and $\xi$. It is shown that if $p(x,t)>\frac{2(N+1)}{N+2}$, $f\in L^2(Q_T)$, $|\nabla u_0|^{p(x,0)}\in L^1(\Omega)$, then the problem has a solution $u\in C^0([0,T];L^2(\Omega))$ with $|\nabla u|^{p(x,t)} \in L^{\infty}(0,T;L^1(\Omega))$, $u_t\in L^2(Q_T)$, obtained as the limit of solutions to the regularized problems in the parabolic H\"older space. The solution possesses the following global regularity properties: \[ \begin{split} & |\nabla u|^{2(p(x,t)-1)+r}\in L^1(Q_T)\quad \text{for any $0 < r < \frac{4}{N+2}$}, \\ & |\nabla u|^{p(x,t)-2} \nabla u \in W^{1,2}(Q_T)^N. \end{split} \]