Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

TL;DR

This paper establishes existence, uniqueness, and regularity results for a class of parabolic equations with variable growth and double phase flux, using new interpolation inequalities in variable Sobolev spaces.

Contribution

It introduces novel regularity and existence results for parabolic problems with variable exponents and double phase flux, including higher integrability and second-order differentiability.

Findings

Existence of unique strong solutions under specified conditions.

Global higher integrability of the gradient with variable exponents.

Second-order differentiability of solutions proven.

Abstract

We study the homogeneous Dirichlet problem for the equation \[ u_t-\operatorname{div}\left((a(z)\vert \nabla u\vert ^{p(z)-2}+b(z)\vert \nabla u\vert ^{q(z)-2})\nabla u\right)=f\quad \text{in $Q_T=\Omega\times (0,T)$}, \] where $\Omega\subset \mathbb{R}^N$, $N\geq 2$, is a bounded domain with $\partial\Omega \in C^2$. The variable exponents $p$, $q$ and the nonnegative modulating coefficients $a$, $b$ are given Lipschitz-continuous functions of the argument $z=(x,t)\in Q_T$. It is assumed that $\frac{2N}{N+2}<p(z),\ q(z)$ and that the modulating coefficients and growth exponents satisfy the balance conditions \[ \text{$a(z)+b(z)\geq \alpha>0$ in $\overline{Q}_T$},\; \alpha=const;\qquad \text{$\vert p(z)-q(z)\vert <\frac{2}{N+2}$ in $\overline{Q}_T$}. \] We find conditions on the source $f$ and the initial data $u(\cdot,0)$ that guarantee the existence of a unique strong solution $u$ with $u_t\in L^2(Q_T)$ and $a\vert \nabla u\vert ^{p}+b\vert \nabla u\vert ^q\in L^\infty(0,T;L^1(\Omega))$. The solution possesses the property of global higher integrability of the gradient, \[ \vert \nabla u\vert ^{\min\{p(z),q(z)\}+r}\in L^1(Q_T)\quad \text{with any $r\in \left(0,\frac{4}{N+2}\right)$}, \] which is derived with the help of new interpolation inequalities in the variable Sobolev spaces. The second-order differentiability of the strong solution is proven: \[ D_{x_i}\left(\left(a\vert \nabla u\vert ^{p-2}+b\vert \nabla u\vert ^{q-2}\right)^{\frac{1}{2}}D_{x_j}u\right)\in L^2(Q_T),\quad i,j=1,2,\ldots,N. \]