Absence of dead-core formations in chemotaxis systems with degenerate diffusion
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909380219240448 |
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| author | Black, Tobias |
| author_facet | Black, Tobias |
| contents | In this paper we consider a chemotaxis system with signal consumption and degenerate diffusion of the form \begin{align*} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\nabla\cdot\big(D(u)\nabla u-uS(u)\nabla v\big)+f(u,v),\\ &v_t=Δv- uv,\\ \end{array}\right. \end{align*} in a bounded domain $Ω\subset\mathbb{R}^{N}$ with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient $D\in C^0([0,\infty))\cap C^2((0,\infty))$ is assumed to satisfy $D(0)=0$, $D(s)>0$ on $(0,\infty)$, $D'(s)\geq 0$ on $(0,\infty)$ and that there are $s_0>0$, $p>1$ and $C_D>0$ such that $$s D'(s)\leq C_D D(s)\quad\text{and}\quad C_D s^{p-1}\leq D(s)\quad\text{for }s\in[0,s_0].$$ The sensitivity function $S\in C^2([0,\infty))$ and the source term $f\in C^{1}([0,\infty)\times[0,\infty))$ are supposed to be nonnegative.
We show that for all suitably regular initial data $(u_0,v_0)$ satisfying $u_0\geq δ_0>0$ and $v_0\not\equiv 0$ there is a time-local classical solution and - despite the degeneracy at $0$ - the solution satisfies an extensibility criterion of the form $$\text{either}\quad T_{max}=\infty,\quad\text{or}\quad\limsup_{t\nearrow T_{max}}\|u(\cdot,t)\|_{L^\infty(Ω)}=\infty.$$ Moreover, as a by-product of our analysis, we prove that a classical solution on $Ω\times(0,T)$ obeying $\|u(\cdot,t)\|_{L^\infty(Ω)}\leq M_u$ for all $t\in(0,T)$ and emanating from initial data $(u_0,v_0)$ as specified above remains strictly positive throughout $Ω\times(0,T)$, i.e. one can find $δ_u=δ_u(T,δ_0, M_u,\|v_0\|_{W^{1,\infty}(Ω)})>0$ such that $$u(x,t)\geqδ_u\quad\text{for all }(x,t)\inΩ\times(0,T).$$ Together, the results indicate that the formation of a dead-core in these chemotaxis systems with a degenerate diffusion are impossible before the blow-up time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14890 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Absence of dead-core formations in chemotaxis systems with degenerate diffusion Black, Tobias Analysis of PDEs 35A01, 35B09, 35K65, 35Q92, 92C17 In this paper we consider a chemotaxis system with signal consumption and degenerate diffusion of the form \begin{align*} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\nabla\cdot\big(D(u)\nabla u-uS(u)\nabla v\big)+f(u,v),\\ &v_t=Δv- uv,\\ \end{array}\right. \end{align*} in a bounded domain $Ω\subset\mathbb{R}^{N}$ with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient $D\in C^0([0,\infty))\cap C^2((0,\infty))$ is assumed to satisfy $D(0)=0$, $D(s)>0$ on $(0,\infty)$, $D'(s)\geq 0$ on $(0,\infty)$ and that there are $s_0>0$, $p>1$ and $C_D>0$ such that $$s D'(s)\leq C_D D(s)\quad\text{and}\quad C_D s^{p-1}\leq D(s)\quad\text{for }s\in[0,s_0].$$ The sensitivity function $S\in C^2([0,\infty))$ and the source term $f\in C^{1}([0,\infty)\times[0,\infty))$ are supposed to be nonnegative. We show that for all suitably regular initial data $(u_0,v_0)$ satisfying $u_0\geq δ_0>0$ and $v_0\not\equiv 0$ there is a time-local classical solution and - despite the degeneracy at $0$ - the solution satisfies an extensibility criterion of the form $$\text{either}\quad T_{max}=\infty,\quad\text{or}\quad\limsup_{t\nearrow T_{max}}\|u(\cdot,t)\|_{L^\infty(Ω)}=\infty.$$ Moreover, as a by-product of our analysis, we prove that a classical solution on $Ω\times(0,T)$ obeying $\|u(\cdot,t)\|_{L^\infty(Ω)}\leq M_u$ for all $t\in(0,T)$ and emanating from initial data $(u_0,v_0)$ as specified above remains strictly positive throughout $Ω\times(0,T)$, i.e. one can find $δ_u=δ_u(T,δ_0, M_u,\|v_0\|_{W^{1,\infty}(Ω)})>0$ such that $$u(x,t)\geqδ_u\quad\text{for all }(x,t)\inΩ\times(0,T).$$ Together, the results indicate that the formation of a dead-core in these chemotaxis systems with a degenerate diffusion are impossible before the blow-up time. |
| title | Absence of dead-core formations in chemotaxis systems with degenerate diffusion |
| topic | Analysis of PDEs 35A01, 35B09, 35K65, 35Q92, 92C17 |
| url | https://arxiv.org/abs/2409.14890 |