Boundedness in a two-dimensional doubly degenerate nutrient taxis system

In this work, we study the doubly degenerate nutrient taxis system [Formula: see text] in a smoothly bounded domain [Formula: see text], where [Formula: see text] and [Formula: see text]. This model was proposed by [J. F. Leyva, C. Málaga and R. G. Plaza, The effects of nutrient chemotaxis on bacterial aggregation patterns with nonlinear degenerate cross diffusion, Phys. A 392 (2013) 5644–5662] to describe the experimentally observed complex pattern formation phenomena in bacterial populations. In this paper, we demonstrate that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem ( * ) possesses a global bounded weak solution, which is continuous in its first component and essentially smooth in its second component. Compared to the existing works, we do not impose any smallness conditions on the initial data. Moreover, we identify a criterion regarding the initial smallness of the second component such that the solution stabilizes to a nonconstant steady state.