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arXiv 2407.20133math.AP
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具有非标准增长的抛物型方程解的全局梯度估计

Global gradient estimates for solutions of parabolic equations with nonstandard growth

Rakesh Arora, Sergey Shmarev

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30 pages. Comments are welcome
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AI总结：

研究初始数据和平滑项光滑性对 $p(x,t)$-Laplace 型抛物型方程 Dirichlet 问题解全局正则性的影响，证明了解的全局高阶可积性与二阶正则性，并给出了分部积分公式。

摘要：

我们研究初始数据和平滑项的光滑性如何影响 $p(x,t)$-Laplace 型抛物型方程类 Dirichlet 问题解的全局正则性性质：\\[ u_t-Δ_{p(\cdot)}u=f(z)+F(z,u,\nabla u),\quad z=(x,t)\in Q_T=Ω\times (0,T), \\] 其中非线性源项 $F(z,u,\nabla u)=a(z)|u|^{q(z)-2}u+|\nabla u|^{s(z)-2}(\vec c,\nabla u)$。证明了存在这样的解：若对于某些 $r\geq \max\{2,\max p(z)\}$ 满足 $|\nabla u(x,0)|\in L^r(Ω)$，则梯度在时间上保持初始的可积阶数，获得全局高阶可积性，并且解在以下意义下获得二阶正则性：\\[ \text{$|\nabla u(x,t)|\in L^r(Ω)$ 对几乎所有 $t \in (0,T)$}, \qquad \text{$|\nabla u|^{p(z)+ρ+r-2} \in L^1(Q_T)$ 对任意 $ρ\in \left(0, \frac{4}{N+2}\right)$}, \\] 以及 \\[ |\nabla u|^{\frac{p(z)+r}{2}-2}\nabla u\in L^2(0,T;W^{1,2}(Ω))^N. \\] 若 $f\in L^{N+2}(Q_T)$，指数 $r$ 是任意且独立于 $p(z)$ 的；而对于 $f\in L^σ(Q_T)$ 且 $σ\in (2,N+2)$ 的情况，指数 $r$ 属于一个有界区间，其端点由 $\max p(z)$、$\min p(z)$、$N$ 和 $σ$ 定义。此外还证明了一个分部积分公式，该公式具有独立意义。

英文摘要：

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: \[ u_t-Δ_{p(\cdot)}u=f(z)+F(z,u,\nabla u),\quad z=(x,t)\in Q_T=Ω\times (0,T), \] 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 $|\nabla u(x,0)|\in L^r(Ω)$ 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: \[ \text{$|\nabla u(x,t)|\in L^r(Ω)$ for a.e. $t \in (0,T)$}, \qquad \text{$|\nabla u|^{p(z)+ρ+r-2} \in L^1(Q_T)$ for any $ρ\in \left(0, \frac{4}{N+2}\right)$}, \] and \[ |\nabla u|^{\frac{p(z)+r}{2}-2}\nabla u\in L^2(0,T;W^{1,2}(Ω))^N. \] The exponent $r$ is arbitrary and independent of $p(z)$ if $f\in L^{N+2}(Q_T)$, while for $f\in L^σ(Q_T)$ with $σ\in (2,N+2)$ the exponent $r$ belongs to a bounded interval whose endpoints are defined by $\max p(z)$, $\min p(z)$, $N$, and $σ$. An integration by parts formula is also proven, which is of independent interest.

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