Uniqueness and large-time continuous dependence in a doubly degenerate nutrient taxis system

Abstract

\[ u_t = \nabla \cdot (uv\nabla u) - χ\nabla\cdot (u^2 v\nabla v) + \ell uv, \qquad \qquad v_t = Δv - uv, \] is considered in a smoothly bounded domain $Ω\subset {\mathbb{R}}^n$ and with $χ\ge 0$ and $\ell\ge 0$. Within a suitable concept of solvability, a statement on uniqueness of solutions emanating from initial data $(u_0,v_0)$ is derived under the mere assumption that $u_0\in L^\infty(Ω)$ and $v_0\in W^{1,\infty}(Ω)$ are such that $u_0\ge 0$ and $v_0>0$ in $\overlineΩ$; in particular, no condition on continuity of $u_0$ is imposed. This result is thereafter used to reveal that in physcally meaningful scenarios, for any initial data which are additionally assumed to be continuous and nontrivial, such solutions can always be found, and that these solutions depend continuously on the initial distributions not only in bounded sets of the form $Ω\times (0,T)$, but actually throughout $Ω\times (0,\infty)$, in a suitable sense.