Weak global solvability of a doubly degenerate parabolic-elliptic nutrient taxis system

Abstract

This work studies the following doubly degenerate parabolic-elliptic nutrient taxis system $$\{\begin{array}{l}{u}_t=(uv{u}_x{)}_x-(u^{2}v{v}_x{)}_x+uv,\\ 0=v_{xx}-uv+f(x,t),\end{array}$$ in a bounded interval $\Omega \subset \mathbb{R}$, under no-flux boundary conditions and nonnegative initial value $u(x,0)=u_0(x)\ge 0$, where $f(x,t)\ge 0$ is known external supply of the nutrient. It is shown that for any nonnegative $u_0\in W^{1,\infty }(\Omega )$ and $f\in C^1(\bar{\Omega }\times [0,\infty ))$, $f\not\equiv 0$, a global weak solution of the problem can be constructed by means of a regularization approach. The core of the analysis lies on a Harnack-type inequality for the second that allows us to overcome the lack of uniform coercivity. Together with time regularity properties, we obtain relative compactness through a combination of the Arzelà-Ascoli theorem and the Aubin-Lions lemma.