Nonnegative solutions to a doubly degenerate nutrient taxis system

This paper deals with the doubly degenerate nutrient taxis system document eqnarray* \ array{ll uₜ = (uv uₓ) ₓ - (u² vvₓ) ₓ + uv, x, \ t0, \\ vₜ = vₗₗ -uv, x, \ t0, array. eqnarray* document in an open bounded interval document R document, with document 0 document, which has been proposed to model the formation of diverse morphological aggregation patterns observed in colonies of Bacillus subtilis growing on the surface of thin agar plates. It is shown that under the mere assumption that document eqnarray* \ array{l u₀ W^1, () is nonnegative with u₀ 0 and \\ v₀ W^1, () is positive in, array. () eqnarray* document an associated no-flux initial boundary value problem possesses a globally defined and continuous weak solution document (u, v) document, where document u 0 document and document v0 document in document 0, ) document, and that moreover there exists document u_ C⁰ () document such that the solution document (u, v) document approaches the pair document (u_, 0) document in the large time limit with respect to the topology document (L^ () ) ² document. This extends comparable results recently obtained in [17, the latter crucially relying on the additional requirement that document _ u₀- document, to situations involving nontrivially supported initial data document u₀ document, which seems to be of particular relevance in the addressed application context of sparsely distributed populations.