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

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 $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$, 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 Hölder space. The solution possesses the following global regularity properties: \[ \begin{split} & |\nabla u|^{2(p(x,t)-1)+r}\in L^1(Q_T)\quad \text{for any $0<r<\frac{4}{N+2}$}, \\ & |\nabla u|^{p(x,t)-2} \nabla u \in W^{1,2}(Q_T)^N. \end{split} \]