Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.10065 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913501562273792 |
|---|---|
| author | Bezerra, Flank D. M. Sastre-Gomez, Silvia da Silva, Severino H. |
| author_facet | Bezerra, Flank D. M. Sastre-Gomez, Silvia da Silva, Severino H. |
| contents | In this paper we consider the following nonlocal autonomous evolution equation in a bounded domain $Ω$ in $\mathbb{R}^N$ \[ \partial_t u(x,t) =- h(x)u(x,t) + g \Big(\int_Ω J(x,y)u(y,t)dy \Big) +f(x,u(x,t)) \] where $h\in W^{1,\infty}(Ω)$, $g: \mathbb{R} \to \mathbb{R}$ and $f:\mathbb{R}^N\times\mathbb{R} \to \mathbb{R}$ are continuously differentiable function, and $J$ is a symmetric kernel; that is, $J(x,y)=J(y,x)$ for any $x,y\in\mathbb{R}^N$. Under additional suitable assumptions on $f$ and $g$, we study the asymptotic dynamics of the initial value problem associated to this equation in a suitable phase spaces. More precisely, we prove the existence, and upper semicontinuity of compact global attractors with respect to kernel $J$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10065 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper semicontinuity for a class of nonlocal evolution equations with Neumann condition Bezerra, Flank D. M. Sastre-Gomez, Silvia da Silva, Severino H. Analysis of PDEs In this paper we consider the following nonlocal autonomous evolution equation in a bounded domain $Ω$ in $\mathbb{R}^N$ \[ \partial_t u(x,t) =- h(x)u(x,t) + g \Big(\int_Ω J(x,y)u(y,t)dy \Big) +f(x,u(x,t)) \] where $h\in W^{1,\infty}(Ω)$, $g: \mathbb{R} \to \mathbb{R}$ and $f:\mathbb{R}^N\times\mathbb{R} \to \mathbb{R}$ are continuously differentiable function, and $J$ is a symmetric kernel; that is, $J(x,y)=J(y,x)$ for any $x,y\in\mathbb{R}^N$. Under additional suitable assumptions on $f$ and $g$, we study the asymptotic dynamics of the initial value problem associated to this equation in a suitable phase spaces. More precisely, we prove the existence, and upper semicontinuity of compact global attractors with respect to kernel $J$. |
| title | Upper semicontinuity for a class of nonlocal evolution equations with Neumann condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.10065 |