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

We consider the homogeneous Dirichlet problem for the parabolic equation $$u_t-\mathrm{div}\left(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(x,t)-2}\mathrm{\nabla }u\right)=f(x,t)+F(x,t,u,\mathrm{\nabla }u)$$ in the cylinder $Q_T:=\mathrm{\Omega }\times (0,T)$, where $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 2$, is a $C^2$-smooth or convex bounded domain. It is assumed that $p\in C^{0,1}({\bar{Q}}_T)$ 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 $p(x,t)>\frac{2(N+1)}{N+2}$, $f\in L^2(Q_T)$, $\mathrm{\mid }\mathrm{\nabla }{u}_0{\mathrm{\mid }}^{p(x,0)}\in L^1(\mathrm{\Omega })$, then the problem has a solution $u\in C^0([0,T];L^2(\mathrm{\Omega }))$ with $\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(x,t)}\in L^{\mathrm{\infty }}(0,T;L^1(\mathrm{\Omega }))$, $u_t\in L^2(Q_T)$, obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties: $$\begin{array}{rl}{\displaystyle }& {\displaystyle \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2(p(x,t)-1)+r}\in L^1(Q_T)\text{for any }0<r<\frac{4}{N+2},}\\ {\displaystyle }& {\displaystyle \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(x,t)-2}\mathrm{\nabla }u\in W^{1,2}(Q_T{)}^N\mathrm{.}}\end{array}$$