Nonnegative solutions to a doubly degenerate nutrient taxis system

This paper deals with the doubly degenerate nutrient taxis system

$ \begin{eqnarray*} \left\{ \begin{array}{ll} u_t = (uv u_x)_x - (u^2 vv_x)_x + \ell uv, \qquad & x\in \Omega, \ t>0, \\ v_t = v_{xx} -uv, \qquad & x\in \Omega, \ t>0, \end{array} \right. \end{eqnarray*} $

in an open bounded interval $ \Omega\subset \mathbb{R} $, with $ \ell \ge0 $, 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

$ \begin{eqnarray*} \left\{ \begin{array}{l} u_0\in W^{1,\infty}( \Omega) \mbox{ is nonnegative with } u_0\not\equiv 0 \qquad \mbox{and} \\ v_0\in W^{1,\infty}( \Omega) \mbox{ is positive in } \overline{\Omega}, \end{array} \right. \qquad \qquad (\star) \end{eqnarray*} $

an associated no-flux initial boundary value problem possesses a globally defined and continuous weak solution $ (u,v) $, where $ u\ge 0 $ and $ v>0 $ in $ \overline{\Omega}\times [0,\infty) $, and that moreover there exists $ u_\infty\in C^0( \overline{\Omega}) $ such that the solution $ (u,v) $ approaches the pair $ (u_\infty,0) $ in the large time limit with respect to the topology $ (L^{\infty}( \Omega)) ^2 $. This extends comparable results recently obtained in [17], the latter crucially relying on the additional requirement that $ \int_\Omega \ln u_0>-\infty $, to situations involving nontrivially supported initial data $ u_0 $, which seems to be of particular relevance in the addressed application context of sparsely distributed populations.