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

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: \[ D_{x_i}\left(\left(a\vert \nabla u\vert ^{p-2}+b\vert \nabla u\vert ^{q-2}\right)^{\frac{1}{2}}D_{x_j}u\right)\in L^2(Q_T),\quad i,j=1,2,\ldots,N. \]