Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption

Abstract

This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \begin{eqnarray*} \begin{array}{llc} u_t=Δu-χ\nabla\cdot (u\nabla v)+κu-μu^2,\\ v_t=Δv-uv, \end{array} \end{eqnarray*} in $N$-dimensional bounded smooth domains for suitably regular positive initial data. We shall establish the existence of a global bounded classical solution for suitably large $\mu$ and prove that for any $\mu >0$ there exists a weak solution. Moreover, in the case of $\kappa >0$ convergence to the constant equilibrium $(\frac{\kappa }{\mu },0)$ is shown.

Cited by in corpus (1)

• Boundedness in a fully parabolic chemotaxis system with nonlinear diffusion and sensitivity, and logistic source