Stability of stationary solutions to a multidimensional parabolic-parabolic chemotaxis-consumption model

We study the dynamical stability of nontrivial steady states to a multidimensional parabolic-parabolic chemotaxis-consumption model in a bounded domain, where the physical zero-flux boundary condition for the bacteria and the nonhomogeneous Dirichlet boundary condition for the oxygen are prescribed. We first prove that the spectrum set of the linearized operator at the steady state only consists of eigenvalues with finite algebraic multiplicity. Then we show that all eigenvalues have negative real part if the concentration of the oxygen at boundary is small, or the diffusive coefficient epsilon of the oxygen is large, or epsilon is small. In the radial setting, we further show that all eigenvalues have negative real part without any assumption on the parameters. This result of spectral analysis implies the local asymptotic stability of the nontrivial steady states to the chemotaxis-consumption model under the above assumptions.