Weak solutions to a class of signal-dependent motility Keller-Segel systems with superlinear damping
The existence of global non-negative weak solutions to the Keller-Segel chemotaxis system with logistic term \begin{align*} \partial_t u & = \mathrm{div}(D(v) \nabla u - u S(v)\nabla v) + f(u) \text{ in } (0,\infty)\times\Omega, \\ \partial_t v & = \Delta v - v + u\text{ in } (0,\infty)\times\Omega, \end{align*} supplemented with homogeneous Neumann boundary conditions and initial conditions \(u_0\in C(\bar{\Omega})\), \(u_0\ge 0\), \(u_0\not\equiv 0\), and \(v_0\in W^{2,\infty}(\Omega)\), \(v_0>0\), is investigated. Here, \(\Omega\) is a bounded domain of \(\mathbb{R}^n\), \(n\ge 1\), with smooth boundary, the functions \(D\) and \(S\) belong to \(C^3([0,\infty))\) and satisfy \[ 0 < k \le D(s) \le K\,, \quad |S(s)|\le K\,, \qquad s\ge x_0\,, \] for some \(x_0>0\), and the logistic term \(f\in C^1([0,\infty))\) is such that \(f(0)=0\) and \(f(s) s^{-l} \to -\infty\) as \(s\to\infty\) for some \(l>\max\left\{ \frac{2(n+2)}{n+4} , \frac{2(n-1)}{n} \right\}\). The proof is performed by a compactness approach, the key tool being the use of the superlinear dissipation induced by the logistic term to derive suitable estimates on \(v\) and then on \(u\).
scientific article; zbMATH DE number 7460670
scientific article