Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow

Abstract:

This work is concerned with the doubly degenerate cross-diffusion system \begin{equation}\tag {*} \left \{ \begin {array}{l} u_t = (uvu_x)_x - (u^2 vv_x)_x + uv, \\[1mm] v_t = v_{xx}-uv, \end{array} \right . \end{equation} that has been proposed as a model for experimentally observable quite complex pattern formation phenomena in bacterial populations.

It is shown that for any initial data satisfying adequate regularity and positivity assumptions, a no-flux initial-boundary value problem for the above in a bounded real interval possesses a global weak solution which is continuous in its first and essentially smooth in its second component.

This solution is seen to asymptotically stabilize in the sense that \begin{equation}\tag {**} u(\cdot ,t) \to u_\infty \quad \text {and} \quad v(\cdot ,t)\to 0 \qquad \text {as } t\to \infty \end{equation} with some nonnegative $u_\infty \in C^0(\overline {\Omega })$ which can be obtained as the evaluation of a weak solution $z\in C^0(\overline {\Omega }\times [0,1])$ to a porous medium-type parabolic problem at the finite time $1$.

It is, moreover, revealed that for each suitably regular nonnegative function $u_\star$ on $\Omega$, the pair $(u_\star ,0)$, formally constituting an equilibrium of \eqref{01}, is stable in an appropriate sense. This finally implies a sufficient criterion for the limit $u_\infty$ in \eqref{02} to be spatially heterogeneous.

The latter properties are in sharp contrast to known asymptotic features of corresponding nutrient taxis systems involving linear nondegenerate diffusion, as for which the literature appears to exclusively provide results on solutions which approach spatially constant states in the large time limit.