正交各向异性、广泛退化、双重非线性扩散方程的有界性与压缩估计

中文摘要 AI 辅助

我们研究如下形式的双重非线性正交各向异性发展方程弱解的正则性：\\[ \partial_{t}(\vert u\vert^{\alpha-1}u)-\sum_{i=1}^{N}\partial_{i}\left[a_{i}(x,t)\\,(|\partial_{i}u|-\delta_{i})_{+}^{p-1}\frac{\partial_{i}u}{\vert\partial_{i}u\vert}\right]=f\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\mathrm{in}\\,\\,\\,\Omega_{T}=\Omega\times(0,T), \\] 其中 $\Omega$ 是 $\mathbb{R}^{N}$（$N\geq2$）中的有界开子集，系数 $a_{i}$ 可测且有界，$\alpha>0$，$p\in(1,\infty)$，$\delta_{1},\ldots,\delta_{N}$ 为非负数。我们证明，由于弱解属于适当的 De Giorgi 型能量类，它们是局部有界的。我们还获得了与上述偏微分方程相关的 Cauchy 问题解在空间中的压缩估计和全局有界性。我们的分析推广了文献中已有的关于扩散方程的类似结果，这些方程要么不具有双重非线性，要么退化程度低于本文所考虑的方程。本文的另一个主要新颖之处在于方程右侧存在源项 $f$，我们对其在时空变量中施加了适当的可积性假设。

英文摘要

We study the regularity of weak solutions to doubly nonlinear orthotropic evolution equations of the form \[ \partial_{t}(\vert u\vert^{α-1}u)-\sum_{i=1}^{N}\partial_{i}\left[a_{i}(x,t)\,(|\partial_{i}u|-δ_{i})_{+}^{p-1}\frac{\partial_{i}u}{\vert\partial_{i}u\vert}\right]=f\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω_{T}=Ω\times(0,T), \] where $Ω$ is a bounded open subset of $\mathbb{R}^{N}$ for $N\geq2$, the coefficients $a_{i}$ are measurable and bounded, $α>0$, $p\in(1,\infty)$ and $δ_{1},\ldots,δ_{N}$ are non-negative numbers. We show that weak solutions are locally bounded due to their membership in a suitable De Giorgi-type energy class. We also obtain contractive estimates and global boundedness in space for solutions to a Cauchy problem associated with the above PDE. Our analysis extends analogous results available in the literature for diffusion equations that either do not exhibit double nonlinearity or are less degenerate than those considered here. Another main novelty of this paper is the presence of a source term $f$ on the right-hand side of the equation, for which we impose suitable integrability assumptions in the space-time variables.

发表机构

• Uppsala University（乌普萨拉大学）

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