Immediate smoothing and global solutions for initial data in \(L^1\times W^{1,2}\) in a Keller-Segel system with logistic terms in 2D. (English) Zbl 1469.35075
Summary: This paper deals with the logistic Keller-Segel model
\[
\begin{cases}
u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\kappa u-\mu u^2, \\
v_t=\Delta v-v+u
\end{cases}
\]
in bounded two-dimensional domains (with homogeneous Neumann boundary conditions and for parameters \(\chi,\kappa\in\mathbb{R}\) and \(\mu>0)\), and shows that any nonnegative initial data \((u_0, v_0)\in L^1\times W^{1,2}\) lead to global solutions that are smooth in \(\bar{\Omega}\times (0,\infty)\).
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K59 | Quasilinear parabolic equations |
| 35A09 | Classical solutions to PDEs |
| 92C17 | Cell movement (chemotaxis, etc.) |
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