Uniqueness and nonlinear stability of positive entire solutions in parabolic-parabolic chemotaxis models with logistic source on bounded heterogeneous environments
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914464273530880 |
|---|---|
| author | Issa, Tahir Bachar |
| author_facet | Issa, Tahir Bachar |
| contents | This paper studies the asymptotic behavior of solutions of the parabolic-parabolic chemotaxis model with logistic-type sources in heterogeneous bounded domains: \begin{equation*}
\label{u-v-eq00} \begin{cases} u_t=Δu-χ\nabla\cdot (u \nabla v)+u\Big(a_0(t,x)-a_1(t,x)u-a_2(t,x)\int_Ωu\Big),\quad x\in Ω\cr τv_t=Δv-λv +μu,\quad x\in Ω\cr \frac{\p u}{\p ν}=\frac{\p v}{\p ν}=0,\quad x\in\pΩ. \end{cases}\qquad(\ast)
\end{equation*} \noindent We find parameter regions in which the system has a unique positive entire solution, which is globally asymptotically stable. {More precisely under suitable assumptions on the model's parameters, the system has a unique entire positive solution $(u^*(x,t),v^*(x,t))$ such that for any %$t_0\in\RR$ and $u_0 \in C^0(\barΩ),$ $v_0 \in W^{1,\infty}(\barΩ)$ with $u_0,v_0\ge 0$ and $u_0\not\equiv 0$, the global classical solution $(u(x,t;t_0,u_0,v_0)$, $v(x,t;t_0,u_0,v_0))$ of $(\ast)$ satisfies $$
\lim_{t \to \infty}\Big(\sup_{t_0 \in \mathbb{R}}\|u(\cdot,t;t_0,u_0,v_0)-u^*(\cdot,t)\|_{C^0(\barΩ)}+\sup_{t_0 \in \mathbb{R}}\|v(\cdot,t;t_0,u_0,v_0)-v^*(\cdot,t)\|_{C^0(\barΩ)}\Big)=0. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09311 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness and nonlinear stability of positive entire solutions in parabolic-parabolic chemotaxis models with logistic source on bounded heterogeneous environments Issa, Tahir Bachar Analysis of PDEs 35A09, 35B08, 35B09, 35B40 This paper studies the asymptotic behavior of solutions of the parabolic-parabolic chemotaxis model with logistic-type sources in heterogeneous bounded domains: \begin{equation*} \label{u-v-eq00} \begin{cases} u_t=Δu-χ\nabla\cdot (u \nabla v)+u\Big(a_0(t,x)-a_1(t,x)u-a_2(t,x)\int_Ωu\Big),\quad x\in Ω\cr τv_t=Δv-λv +μu,\quad x\in Ω\cr \frac{\p u}{\p ν}=\frac{\p v}{\p ν}=0,\quad x\in\pΩ. \end{cases}\qquad(\ast) \end{equation*} \noindent We find parameter regions in which the system has a unique positive entire solution, which is globally asymptotically stable. {More precisely under suitable assumptions on the model's parameters, the system has a unique entire positive solution $(u^*(x,t),v^*(x,t))$ such that for any %$t_0\in\RR$ and $u_0 \in C^0(\barΩ),$ $v_0 \in W^{1,\infty}(\barΩ)$ with $u_0,v_0\ge 0$ and $u_0\not\equiv 0$, the global classical solution $(u(x,t;t_0,u_0,v_0)$, $v(x,t;t_0,u_0,v_0))$ of $(\ast)$ satisfies $$ \lim_{t \to \infty}\Big(\sup_{t_0 \in \mathbb{R}}\|u(\cdot,t;t_0,u_0,v_0)-u^*(\cdot,t)\|_{C^0(\barΩ)}+\sup_{t_0 \in \mathbb{R}}\|v(\cdot,t;t_0,u_0,v_0)-v^*(\cdot,t)\|_{C^0(\barΩ)}\Big)=0. $$ |
| title | Uniqueness and nonlinear stability of positive entire solutions in parabolic-parabolic chemotaxis models with logistic source on bounded heterogeneous environments |
| topic | Analysis of PDEs 35A09, 35B08, 35B09, 35B40 |
| url | https://arxiv.org/abs/2401.09311 |