Semilinear Equations Including the Mixed Operator
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910841199132672 |
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| author | Ayoub, Alaa |
| author_facet | Ayoub, Alaa |
| contents | We study the local and global existence of solutions to a semilinear evolution equation driven by a mixed local-nonlocal operator of the form \( L = -Δ+ (-Δ)^{α/2} \), where \( 0 < α< 2 \). The Cauchy problem under consideration is \begin{equation*} \partial_t u + t^βL u = -h(t) u^p, \quad x \in \mathbb{R}^N, \quad t > 0, \end{equation*} with nonnegative initial data \( u(x, 0) = u_0(x) \). We establish the existence and uniqueness of local solutions in \( L^\infty(\mathbb{R}^N) \) using a contraction mapping argument. Furthermore, we analyze conditions for global existence, proving that solutions remain globally bounded in time under appropriate assumptions on the parameters \( β\), \( p \), and the function \( h(t) \). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16646 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semilinear Equations Including the Mixed Operator Ayoub, Alaa Analysis of PDEs We study the local and global existence of solutions to a semilinear evolution equation driven by a mixed local-nonlocal operator of the form \( L = -Δ+ (-Δ)^{α/2} \), where \( 0 < α< 2 \). The Cauchy problem under consideration is \begin{equation*} \partial_t u + t^βL u = -h(t) u^p, \quad x \in \mathbb{R}^N, \quad t > 0, \end{equation*} with nonnegative initial data \( u(x, 0) = u_0(x) \). We establish the existence and uniqueness of local solutions in \( L^\infty(\mathbb{R}^N) \) using a contraction mapping argument. Furthermore, we analyze conditions for global existence, proving that solutions remain globally bounded in time under appropriate assumptions on the parameters \( β\), \( p \), and the function \( h(t) \). |
| title | Semilinear Equations Including the Mixed Operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.16646 |