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Existence of solutions to chemotaxis dynamics with logistic source. (English) Zbl 1354.37098
Summary: This paper is concerned with a chemotaxis system with nonlinear diffusion and logistic growth term \(f(b) = \kappa b-\mu |b|^{\alpha-1}b\) with \(\kappa>0\), \(\mu>0\) and \(\alpha > 1\) under the no-flux boundary condition. It is shown that there exists a local solution to this system for any \(L^2\)-initial data and that under a stronger assumption on the chemotactic sensitivity there exists a global solution for any \(L^2\)-initial data. The proof is based on the method built by G. Marinoschi [J. Math. Anal. Appl. 402, No. 2, 415–439 (2013; Zbl 1272.35120)].

MSC:
37N25 Dynamical systems in biology
92C17 Cell movement (chemotaxis, etc.)
35D30 Weak solutions to PDEs
35M33 Initial-boundary value problems for mixed-type systems of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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