Superlinear degradation in a doubly degenerate nutrient taxis system

Abstract

This work is concerned with the doubly degenerate cross-diffusion system with generalized logistic source ut=∇⋅(uv∇u)−∇⋅(u2v∇v)+f(u),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0in a smoothly bounded domain Ω⊂Rn(n≥2) with no-flux boundary conditions, where f(u)=ρu−μuκ with ρ,μ>0. It is shown that under the assumption that u0∈W1,∞(Ω) is nonnegative with u0⁄≡0 and v0∈W1,∞(Ω) is positive in Ω̄, if κ>(n+2)/2,then the problem admits a global weak solution.