On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics
The authors study the two-species chemotaxis system with Lotka-Volterra competition \[ \begin{cases} \partial_t u = \Delta u - \chi_1 \nabla \cdot (u \nabla w) + \mu_1 u (1-u- a_1 v), & \quad (x,t) \in \Omega \times (0,\infty),\\ \partial_t v = \Delta v - \chi_2 \nabla \cdot (v \nabla w) + \mu_2 v (1-v- a_2 u), & \quad (x,t) \in \Omega \times (0,\infty), \\ \partial_t w = \Delta w - \lambda w + b_1 u + b_2 v, & \quad (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions and sufficiently regular inital data which are nonnegative and nonzero. Here $\Omega \subset \mathbb{R}^3$ is a smooth, bounded domain and $\chi_i, \mu_i, a_i, b_i, \lambda$ are positive constants. $u$ and $v$ denote the densities of two populations which compete for resources and move toward increasing concentrations of the chemoattractant $w$ which they produce themselves. \par The authors prove the existence of a global and bounded classical solution if $\frac{\mu_i}{b_i}$ is suffiently large as compared to $\chi_1$ and $\chi_2$ for $i=1,2$. \par An important ingredient of the proof is the derivation of a differential inequality which can be seen as an entropy type functional and allows to prove the boundedness of appropriate $L^p$-norms of the solution components.
scientific article; zbMATH DE number 6996196
scientific article