Global Boundedness and Asymptotic Behavior in a Chemotaxis-Consumption Model with Signal-Dependent Motility and Indirect Sensing

We study a chemotaxis model with signal-dependent motility and indirect consumption mechanism, described by a cross-degenerate parabolic system. The motility function $\phi (v)$ is assumed to behave as $v^\alpha$ near $v=0,$ leading to cross-degenerate diffusion. Under the presence of a logistic growth term in the cell dynamics, we establish the existence of globally bounded classical solutions provided the crowding coefficient $b$ is sufficiently large. Moreover, for strongly degenerate cases $(\alpha>1),$ we prove uniform convergence of solutions toward the homogeneous steady state $(1/b,0,1/b)$ as $t→∞.$ Our analysis relies on the construction of a conditional energy functional and temporally averaged estimates, which enable us to overcome the difficulties induced by cross-degeneracy. Numerical simulations illustrate how variations in the parameter $b$ and the growth rate $\delta$ influence pattern formation, highlighting transitions from homogeneity to spatially heterogeneous structures.