Boundedness in a two-dimensional doubly degenerate nutrient taxis system

In this work, we study the no-flux initial-boundary value problem for the doubly degenerate nutrient taxis system {align} {cases}{}{eq 0.1} u_t=∇ ·(u v ∇ u)-χ ∇ ·(u² v ∇ v)+ u v, & x ∈ Ω, t>0, \\ v_t=Δ v-u v, & x ∈ Ω, t>0 {cases} {align} in a smoothly bounded convex domain Ω ⊂ R², where χ>0 and ≥ 0. In this paper, we present that for all reasonably regular initial data, the model {eq 0.1} possesses a global bounded weak solution which is continuous in its first and essentially smooth in its second component. {abstract}