Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators
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| Format: | Preprint |
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2024
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| _version_ | 1866910728115453952 |
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| author | Kumagai, Takashi Wang, Jian Zhang, Meng-ge |
| author_facet | Kumagai, Takashi Wang, Jian Zhang, Meng-ge |
| contents | In this paper, we study the local boundedness of local weak solutions to the following parabolic equation associated with fractional $p$-Laplacian type operators $$ \partial_t u(t,x)-\text{p.v.}\int_{\R^d}|u(t,y)-u(t,x)|^{p-2}(u(t,y)-u(t,x))J(t;x,y)\,dy=0,\quad (t,x)\in \R\times \R^d, $$ where $\text{p.v.}$ means the integral in the principal value sense, $p\in(1,\infty)$ and $J(t;x,y)$ is comparable to the kernel of the fractional $p$-Laplacian operator $|x-y|^{-d-sp}$ with $s\in(0,1)$ and uniformly in $(t;x,y)\in\R\times\R^d\times\R^d$. Unlike existing results in the literature, the local boundedness of the solutions obtained in this paper extends the known results for the linear case (i.e., the case that $p=2$), in particular with a nonlocal parabolic tail that uses the $L^1$-norm in time for all $p\in (1,\infty)$. The proof is based on a new level set truncation in the De Giorgi-Nash-Moser iteration and a careful choice of iteration orders, as well as a general Caccioppoli-type inequality that is efficiently applied to fractional $p$-Laplacian type operators with all $p>1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03770 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators Kumagai, Takashi Wang, Jian Zhang, Meng-ge Analysis of PDEs In this paper, we study the local boundedness of local weak solutions to the following parabolic equation associated with fractional $p$-Laplacian type operators $$ \partial_t u(t,x)-\text{p.v.}\int_{\R^d}|u(t,y)-u(t,x)|^{p-2}(u(t,y)-u(t,x))J(t;x,y)\,dy=0,\quad (t,x)\in \R\times \R^d, $$ where $\text{p.v.}$ means the integral in the principal value sense, $p\in(1,\infty)$ and $J(t;x,y)$ is comparable to the kernel of the fractional $p$-Laplacian operator $|x-y|^{-d-sp}$ with $s\in(0,1)$ and uniformly in $(t;x,y)\in\R\times\R^d\times\R^d$. Unlike existing results in the literature, the local boundedness of the solutions obtained in this paper extends the known results for the linear case (i.e., the case that $p=2$), in particular with a nonlocal parabolic tail that uses the $L^1$-norm in time for all $p\in (1,\infty)$. The proof is based on a new level set truncation in the De Giorgi-Nash-Moser iteration and a careful choice of iteration orders, as well as a general Caccioppoli-type inequality that is efficiently applied to fractional $p$-Laplacian type operators with all $p>1$. |
| title | Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.03770 |