Regularity for quasi-linear parabolic equations with nonhomogeneous degeneracy or singularity
Abstract
We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $$ \partial_t u=[|D u|^q+a(x,t)|D u|^s]\left(\Delta u+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D u|}\right\rangle\right), $$ where $1<p<\infty$, $-1<q\leq s<\infty$ and $a(x,t)\ge 0$. The motivation to investigate this model stems not only from the connections to tug-of-war like stochastic games with noise, but also from the non-standard growth problems of double phase type. According to different values of $q,s$, such equations include nonhomogeneous degeneracy or singularity, and may involve these two features simultaneously. In particular, when $q=p-2$ and $q<s$, it will encompass the parabolic $p$-Laplacian both in divergence form and in non-divergence form. We aim to explore the from $L^\infty$ to $C^{1,\alpha}$ regularity theory for the aforementioned problem. To be precise, under some proper assumptions, we use geometrical methods to establish the local Hölder regularity of spatial gradients of viscosity solutions.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2021
- DOI:
- 10.48550/arXiv.2105.04743
- arXiv:
- arXiv:2105.04743
- Bibcode:
- 2021arXiv210504743F
- Keywords:
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- Mathematics - Analysis of PDEs