Hölder regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime
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2025
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| author | Adimurthi, Karthik Modasiya, Mitesh |
| author_facet | Adimurthi, Karthik Modasiya, Mitesh |
| contents | We study local Hölder regularity of bounded, weak solutions for the nonlocal quasilinear equations of the form \[ (|u|^{q-2}u)_t + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}} dy = 0, \] with $p\in (1,\infty)$, $q\in (1,\infty)$ and $s \in (0,1)$. Analogous Hölder continuity result in the local case is known in the purely singular case $\{1<p<2, p<q\}$, purely degenerate case $\{2<p, q<p\}$, scale invariant case $\{p=q\}$ and translation invariant case $\{q=2,1<p<\infty\}$. In the nonlocal setting, Hölder regularity is known when the equation is either translation invariant $\{q=2, 1<p<\infty\}$ or scale invariant $\{q=p, 1<p<\infty\}$ or purely degenerate case $\{2<p, q<p\}$. Similar strategy can be used to obtain Hölder regularity in the purely singular case $\{1<p<2, p<q\}$.
In this paper, we adapt several ideas developed over the past few years and combine it with a new intrinsic scaling to prove Hölder regularity in the mixed singular-degenerate range $\max\{p,q,2\} < \min\left\{q + \tfrac{p-1}{1+\frac{n}{sp}}, 2 + \tfrac{p-1}{1+\frac{n}{sp}}\right\}$. The proof explicitly makes use of the nonlocal nature of the problem and as a consequence, our estimates are not stable at $s \rightarrow 0$. We note that the analogous regularity in the local problem remains open. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_19421 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hölder regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime Adimurthi, Karthik Modasiya, Mitesh Analysis of PDEs 35K51, 35A01, 35A15, 35R11 We study local Hölder regularity of bounded, weak solutions for the nonlocal quasilinear equations of the form \[ (|u|^{q-2}u)_t + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}} dy = 0, \] with $p\in (1,\infty)$, $q\in (1,\infty)$ and $s \in (0,1)$. Analogous Hölder continuity result in the local case is known in the purely singular case $\{1<p<2, p<q\}$, purely degenerate case $\{2<p, q<p\}$, scale invariant case $\{p=q\}$ and translation invariant case $\{q=2,1<p<\infty\}$. In the nonlocal setting, Hölder regularity is known when the equation is either translation invariant $\{q=2, 1<p<\infty\}$ or scale invariant $\{q=p, 1<p<\infty\}$ or purely degenerate case $\{2<p, q<p\}$. Similar strategy can be used to obtain Hölder regularity in the purely singular case $\{1<p<2, p<q\}$. In this paper, we adapt several ideas developed over the past few years and combine it with a new intrinsic scaling to prove Hölder regularity in the mixed singular-degenerate range $\max\{p,q,2\} < \min\left\{q + \tfrac{p-1}{1+\frac{n}{sp}}, 2 + \tfrac{p-1}{1+\frac{n}{sp}}\right\}$. The proof explicitly makes use of the nonlocal nature of the problem and as a consequence, our estimates are not stable at $s \rightarrow 0$. We note that the analogous regularity in the local problem remains open. |
| title | Hölder regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime |
| topic | Analysis of PDEs 35K51, 35A01, 35A15, 35R11 |
| url | https://arxiv.org/abs/2512.19421 |