Logistic elliptic and parabolic problem for the fractional $p$-Laplacian
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912816753016832 |
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| author | Constantin, Loïc Santos, Carlos Alberto Warnault, Guillaume |
| author_facet | Constantin, Loïc Santos, Carlos Alberto Warnault, Guillaume |
| contents | In this paper we prove existence, uniqueness of weak solutions of the following nonlocal nonlinear logistic equation \begin{equation*} \begin{cases} (-Δ)_p^s u_λ=λu_λ^q - b(x)u_λ^r \quad \text{in} \;Ω,\\ u_λ=0 \quad \text{in} \; ( \mathbb{R}^d \backslash Ω), \\ u_λ>0 \text{ in} \; Ω. \end{cases}\ \end{equation*} We also prove behavior of $u_λ$ with respect to $λ,$ underlining the effect of the nonlocal operator. We then study the associated parabolic problem, proving local and global existence, uniqueness and global behavior such as stabilization, finite time extinction and blow up. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_23272 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Logistic elliptic and parabolic problem for the fractional $p$-Laplacian Constantin, Loïc Santos, Carlos Alberto Warnault, Guillaume Analysis of PDEs In this paper we prove existence, uniqueness of weak solutions of the following nonlocal nonlinear logistic equation \begin{equation*} \begin{cases} (-Δ)_p^s u_λ=λu_λ^q - b(x)u_λ^r \quad \text{in} \;Ω,\\ u_λ=0 \quad \text{in} \; ( \mathbb{R}^d \backslash Ω), \\ u_λ>0 \text{ in} \; Ω. \end{cases}\ \end{equation*} We also prove behavior of $u_λ$ with respect to $λ,$ underlining the effect of the nonlocal operator. We then study the associated parabolic problem, proving local and global existence, uniqueness and global behavior such as stabilization, finite time extinction and blow up. |
| title | Logistic elliptic and parabolic problem for the fractional $p$-Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.23272 |