Existence and global behaviour of solutions of a parabolic problem involving the fractional $p$-Laplacian in porous medium
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| Format: | Preprint |
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2024
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| _version_ | 1866916490967515136 |
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| author | Constantin, Loïc Giacomoni, Jacques Warnault, Guillaume |
| author_facet | Constantin, Loïc Giacomoni, Jacques Warnault, Guillaume |
| contents | In this paper, we prove the existence and the uniqueness of a weak and mild solution of the following nonlinear parabolic problem involving the porous $p$-fractional Laplacian:
\begin{equation*} \begin{cases} \partial_t u+(-Δ)^s_p(|u|^{m-1}u)=h(t,x,|u|^{m-1}u) & \text{in} \; (0,T)\times Ω,\\ u=0 & \text{in} \; (0,T) \times \mathbb{R}^d\backslash Ω, \\ u(0,\cdot)=u_0 & \text{in} \; Ω. \end{cases}\ \end{equation*} We also study further the the homogeneous case $h(u)=|u|^{q-1}u$ with $q>0$. In particular we investigate global time existence, uniqueness, global behaviour of weak solutions and stabilization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_14260 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and global behaviour of solutions of a parabolic problem involving the fractional $p$-Laplacian in porous medium Constantin, Loïc Giacomoni, Jacques Warnault, Guillaume Analysis of PDEs In this paper, we prove the existence and the uniqueness of a weak and mild solution of the following nonlinear parabolic problem involving the porous $p$-fractional Laplacian: \begin{equation*} \begin{cases} \partial_t u+(-Δ)^s_p(|u|^{m-1}u)=h(t,x,|u|^{m-1}u) & \text{in} \; (0,T)\times Ω,\\ u=0 & \text{in} \; (0,T) \times \mathbb{R}^d\backslash Ω, \\ u(0,\cdot)=u_0 & \text{in} \; Ω. \end{cases}\ \end{equation*} We also study further the the homogeneous case $h(u)=|u|^{q-1}u$ with $q>0$. In particular we investigate global time existence, uniqueness, global behaviour of weak solutions and stabilization. |
| title | Existence and global behaviour of solutions of a parabolic problem involving the fractional $p$-Laplacian in porous medium |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.14260 |