Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source
Abstract In this paper, we study the following chemotaxis system with a signal-dependent motility and logistic source: { u t = Δ ( γ ( v ) u ) + μ u ( 1 − u α ) , x ∈ Ω , t > 0 , 0 = Δ v − v + u r , x ∈ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω $$ \textstyle\begin{cases} u_{t}=\Delta {\bigl(...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-08-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-023-01766-7 |
| _version_ | 1797451934730289152 |
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| author | Yanmei Hu Wanjuan Du |
| author_facet | Yanmei Hu Wanjuan Du |
| author_sort | Yanmei Hu |
| collection | DOAJ |
| description | Abstract In this paper, we study the following chemotaxis system with a signal-dependent motility and logistic source: { u t = Δ ( γ ( v ) u ) + μ u ( 1 − u α ) , x ∈ Ω , t > 0 , 0 = Δ v − v + u r , x ∈ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω $$ \textstyle\begin{cases} u_{t}=\Delta {\bigl(\gamma (v)u\bigr)}+\mu u\bigl(1-u^{\alpha}\bigr), &x \in \Omega , t > 0, \\ 0=\Delta v-\ v+u^{r} , &x\in \Omega , t > 0, \\ u(x, 0) = u_{0}(x), &x\in \Omega \end{cases} $$ under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R 2 $\Omega \subset \mathbb{R}^{2}$ , where the motility function γ ( v ) $\gamma (v)$ satisfies γ ( v ) ∈ C 3 ( [ 0 , ∞ ) ) $\gamma (v)\in C^{3}([0,\infty ))$ with γ ( v ) > 0 $\gamma (v)>0$ , and | γ ′ ( v ) | 2 γ ( v ) $\frac{|\gamma '(v)|^{2}}{\gamma (v)}$ is bounded for all v > 0 $v > 0$ . The purpose of this paper is to prove that the model possesses globally bounded solutions. In addition, we show that all solutions ( u , v ) $(u, v)$ of the model will exponentially converge to the unique constant steady state ( 1 , 1 ) $(1, 1)$ as t → + ∞ $t\rightarrow +\infty $ when μ ≥ K 4 1 + r $\mu \geq \frac{K}{4^{1+r}}$ with K = max 0 < v ≤ ∞ | γ ′ ( v ) | 2 γ ( v ) $K=\max_{0< v\leq \infty} \frac{|\gamma '(v)|^{2}}{\gamma (v)}$ . |
| first_indexed | 2024-03-09T15:01:39Z |
| format | Article |
| id | doaj.art-095d990147cc4613ae60f943f2f4a7c0 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-03-09T15:01:39Z |
| publishDate | 2023-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-095d990147cc4613ae60f943f2f4a7c02023-11-26T13:51:02ZengSpringerOpenBoundary Value Problems1687-27702023-08-012023112210.1186/s13661-023-01766-7Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic sourceYanmei Hu0Wanjuan Du1College of Mathematic and Information, China West Normal UniversityCollege of Mathematics Education, China West Normal UniversityAbstract In this paper, we study the following chemotaxis system with a signal-dependent motility and logistic source: { u t = Δ ( γ ( v ) u ) + μ u ( 1 − u α ) , x ∈ Ω , t > 0 , 0 = Δ v − v + u r , x ∈ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω $$ \textstyle\begin{cases} u_{t}=\Delta {\bigl(\gamma (v)u\bigr)}+\mu u\bigl(1-u^{\alpha}\bigr), &x \in \Omega , t > 0, \\ 0=\Delta v-\ v+u^{r} , &x\in \Omega , t > 0, \\ u(x, 0) = u_{0}(x), &x\in \Omega \end{cases} $$ under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R 2 $\Omega \subset \mathbb{R}^{2}$ , where the motility function γ ( v ) $\gamma (v)$ satisfies γ ( v ) ∈ C 3 ( [ 0 , ∞ ) ) $\gamma (v)\in C^{3}([0,\infty ))$ with γ ( v ) > 0 $\gamma (v)>0$ , and | γ ′ ( v ) | 2 γ ( v ) $\frac{|\gamma '(v)|^{2}}{\gamma (v)}$ is bounded for all v > 0 $v > 0$ . The purpose of this paper is to prove that the model possesses globally bounded solutions. In addition, we show that all solutions ( u , v ) $(u, v)$ of the model will exponentially converge to the unique constant steady state ( 1 , 1 ) $(1, 1)$ as t → + ∞ $t\rightarrow +\infty $ when μ ≥ K 4 1 + r $\mu \geq \frac{K}{4^{1+r}}$ with K = max 0 < v ≤ ∞ | γ ′ ( v ) | 2 γ ( v ) $K=\max_{0< v\leq \infty} \frac{|\gamma '(v)|^{2}}{\gamma (v)}$ .https://doi.org/10.1186/s13661-023-01766-7Global boundednessLogistic sourceChemotaxisSignal-dependent motilityExponential decay |
| spellingShingle | Yanmei Hu Wanjuan Du Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source Boundary Value Problems Global boundedness Logistic source Chemotaxis Signal-dependent motility Exponential decay |
| title | Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source |
| title_full | Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source |
| title_fullStr | Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source |
| title_full_unstemmed | Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source |
| title_short | Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source |
| title_sort | boundedness in a two dimensional chemotaxis system with signal dependent motility and logistic source |
| topic | Global boundedness Logistic source Chemotaxis Signal-dependent motility Exponential decay |
| url | https://doi.org/10.1186/s13661-023-01766-7 |
| work_keys_str_mv | AT yanmeihu boundednessinatwodimensionalchemotaxissystemwithsignaldependentmotilityandlogisticsource AT wanjuandu boundednessinatwodimensionalchemotaxissystemwithsignaldependentmotilityandlogisticsource |