Global gradient estimates for solutions of parabolic equations with nonstandard growth (2407.20133v1)

Abstract: We study how the smoothness of the initial datum and the free term affect the global regularity properties of solutions to the Dirichlet problem for the class of parabolic equations of $p(x,t)$-Laplace type %with nonlinear sources depending on the solution and its gradient: [ u_t-\Delta_{p(\cdot)}u=f(z)+F(z,u,\nabla u),\quad z=(x,t)\in Q_T=\Omega\times (0,T), ] with the nonlinear source $F(z,u,\nabla u)=a(z)|u|^{q(z)-2}u+|\nabla u|^{s(z)-2}(\vec c,\nabla u)$. It is proven the existence of a solution such that if $|\nabla u(x,0)|\in L^r(\Omega)$ for some $r\geq \max{2,\max p(z)}$, then the gradient preserves the initial order of integrability in time, gains global higher integrability, and the solution acquires the second-order regularity in the following sense: [ \text{$|\nabla u(x,t)|\in L^r(\Omega)$ for a.e. $t \in (0,T)$}, \qquad \text{$|\nabla u|^{p(z)+\rho+r-2} \in L^1(Q_T)$ for any $\rho \in \left(0, \frac{4}{N+2}\right)$}, ] and [ |\nabla u|^{\frac{p(z)+r}{2}-2}\nabla u\in L^2(0,T;W^{1,2}(\Omega))^N. ] The exponent $r$ is arbitrary and independent of $p(z)$ if $f\in L^{N+2}(Q_T)$, while for $f\in L^\sigma(Q_T)$ with $\sigma \in (2,N+2)$ the exponent $r$ belongs to a bounded interval whose endpoints are defined by $\max p(z)$, $\min p(z)$, $N$, and $\sigma$. An integration by parts formula is also proven, which is of independent interest.