Boundedness and asymptotic behavior in a fully parabolic chemotaxis-growth system with signal-dependent sensitivity

Abstract

This paper deals with a fully parabolic chemotaxis-growth system with signal-dependent sensitivity $$\left\{\begin{array}{ll}u_t=\Delta u-\nabla\cdot(u\chi(v)\nabla v)+\mu u(1-u), \quad &\quad(x,t)\in\Omega\times (0,\infty),\\ v_{t}=\varepsilon \Delta v+h(u,v), \quad &\quad(x,t)\in \Omega\times (0,\infty),\end{array}\right.$$ under homogeneous Neumann boundary conditions in a bounded domain $${\Omega\subset {\mathbb{R}}^{n} (n\geq1)}$$ with smooth boundary, where $${\varepsilon\in(0,1), \mu>0}$$ , the function $${\chi(v)}$$ 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.