Nonnegative solutions to a doubly degenerate nutrient taxis system

Abstract

<p style='text-indent:20px;'>This paper deals with the doubly degenerate nutrient taxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{ll} u_t = (uv u_x)_x - (u^2 vv_x)_x + \ell uv, \qquad &amp; x\in \Omega, \ t&gt;0, \\ v_t = v_{xx} -uv, \qquad &amp; x\in \Omega, \ t&gt;0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in an open bounded interval <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R} $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ \ell \ge0 $\end{document}</tex-math></inline-formula>, which has been proposed to model the formation of diverse morphological aggregation patterns observed in colonies of <i>Bacillus subtilis</i> growing on the surface of thin agar plates.</p><p style='text-indent:20px;'>It is shown that under the mere assumption that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>an associated no-flux initial boundary value problem possesses a globally defined and continuous weak solution <inline-formula><tex-math id="M3">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ u\ge 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ v&gt;0 $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M6">\begin{document}$ \overline{\Omega}\times [0,\infty) $\end{document}</tex-math></inline-formula>, and that moreover there exists <inline-formula><tex-math id="M7">\begin{document}$ u_\infty\in C^0( \overline{\Omega}) $\end{document}</tex-math></inline-formula> such that the solution <inline-formula><tex-math id="M8">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula> approaches the pair <inline-formula><tex-math id="M9">\begin{document}$ (u_\infty,0) $\end{document}</tex-math></inline-formula> in the large time limit with respect to the topology <inline-formula><tex-math id="M10">\begin{document}$ (L^{\infty}( \Omega)) ^2 $\end{document}</tex-math></inline-formula>. This extends comparable results recently obtained in [<xref ref-type="bibr" rid="b17">17</xref>], the latter crucially relying on the additional requirement that <inline-formula><tex-math id="M11">\begin{document}$ \int_\Omega \ln u_0&gt;-\infty $\end{document}</tex-math></inline-formula>, to situations involving nontrivially supported initial data <inline-formula><tex-math id="M12">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula>, which seems to be of particular relevance in the addressed application context of sparsely distributed populations.</p>