# American Institute of Mathematical Sciences

June  2020, 40(6): 4039-4058. doi: 10.3934/dcds.2020030

## Refined regularity and stabilization properties in a degenerate haptotaxis system

* Corresponding author: Michael Winkler

Received  January 2019 Revised  May 2019 Published  October 2019

We consider the degenerate haptotaxis system
 $\left\{ \begin{array}{l} u_t = (d(x)u)_{xx} - (d(x)uw_x )_x, \\ w_t = -ug(w), \end{array} \right.$
endowed with no-flux boundary conditions in a bounded open interval
 $\Omega \subset \mathbb{R}$
. It was proposed as a basic model for haptotactic migration in heterogeneous environments. If the diffusion is degenerate in the sense that
 $d$
is non-negative, has a non-empty zero set and satisfies
 $\int_\Omega \frac{1}{d} <\infty$
, then it has been shown in [12] under appropriate assumptions on the initial data that the system has a global generalized solution satisfying in particular
 $u(\cdot,t) \rightharpoonup \frac{\mu_\infty}{d}$
weakly in
 $L^1 (\Omega)$
as
 $t \to \infty$
for some positive constant
 $\mu_\infty$
.
We now prove that under the additional restriction
 $\int_\Omega \frac{1}{d^2} <\infty$
we have the strong convergence
 $u(\cdot,t)\to \frac{\mu_\infty}{d}$
in
 $L^p (\Omega)$
as
 $t \to \infty$
for any
 $p \in (1,2)$
. In addition, with the same restriction on
 $d$
we obtain improved regularity properties of
 $u$
, for instance
 $du \in L^\infty ((0,\infty); L^p(\Omega))$
for any
 $p \in (1,\infty)$
.
Citation: Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030
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