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- Publisher
- Springer Journals
- Copyright
- Copyright © 2016 by Springer International Publishing
- Subject
- Mathematics; Analysis
- ISSN
- 1424-3199
- eISSN
- 1424-3202
- DOI
- 10.1007/s00028-016-0344-4
- Publisher site
- See Article on Publisher Site
Abstract
This paper deals with a fully parabolic chemotaxis-growth system with signal-dependent sensitivity $$\left\{\begin{array}{ll}u_t=\Delta u-\nabla\cdot(u\chi(v)\nabla v)+\mu u(1-u), \quad &\quad(x,t)\in\Omega\times (0,\infty),\\ v_{t}=\varepsilon \Delta v+h(u,v), \quad &\quad(x,t)\in \Omega\times (0,\infty),\end{array}\right.$$ u t = Δ u - ∇ · ( u χ ( v ) ∇ v ) + μ u ( 1 - u ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , v t = ε Δ v + h ( u , v ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , under homogeneous Neumann boundary conditions in a bounded domain $${\Omega\subset {\mathbb{R}}^{n} (n\geq1)}$$ Ω ⊂ R n ( n ≥ 1 ) with smooth boundary, where $${\varepsilon\in(0,1), \mu>0}$$ ε ∈ ( 0 , 1 ) , μ > 0 , the function $${\chi(v)}$$ χ ( v ) is the chemotactic sensitivity and h(u,v) denotes the balance between the production and degradation of the chemical signal which depends explicitly on the living organisms. Firstly, by using an iterative method, we derive global existence and uniform boundedness of solutions for this system. Moreover, by relying on an energy approach, the asymptotic stability of constant equilibria is studied. Finally, we shall give an example to illustrate the theoretical results.
Journal
Journal of Evolution Equations – Springer Journals
Published: Jun 6, 2016