Existence of solutions to chemotaxis dynamics with Lipschitz diffusion and superlinear growth

This paper deals with the chemotaxis system with nonlinear diffusion and superlinear growth term f ( b ) = | b | α − 1 b when n ≤ 3 . It is shown that if α ≤ 4 ( n = 1 ) , α < 1 + 4 n ( n = 2 , 3 ) , then there exists a local solution to this system for any large data. In the case of Lipschitz growth, Marinoschi (2013) [12] established the existence of local solutions to this system with sufficiently small initial data and showed that under a stronger assumption on the chemotactic sensitivity there exists a global solution with large initial data. This paper develops the local solvability with Lipschitz growth to the one with superlinear growth and allows the system to have a local solution with large initial data without any stronger assumption. The key to including the superlinear growth lies in the Yosida approximation of f. In order to remove the smallness assumption on the initial data, this paper provides a precise estimate for approximate solutions.