Abstract
The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved. The equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, it extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homogeneous Neumann boundary conditions. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balance or weak cross-diffusion condition. The detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsager’s principle in thermodynamics. Under detailed balance (and without reaction) the entropy is nonincreasing in time, but counter-examples show that the entropy may increase initially if detailed balance does not hold.
Highlights
Shigesada et al [24] suggested in their seminal paper a diffusive Lotka-Volterra system for two competing species, which is able to describe the segregation of the population and to show pattern formation when time increases
We provide for the first time a global existence analysis for an arbitrary number of population species using the entropy method of [15], and we reveal an astonishing relation between the monotonicity of the entropy and the detailed balance condition of an associated Markov chain
A global existence theorem under more general conditions seems not to be available in the literature. We prove such a result and relate a structural condition on the coefficients ai j with Onsager’s principle of thermodynamics
Summary
Shigesada et al [24] suggested in their seminal paper a diffusive Lotka-Volterra system for two competing species, which is able to describe the segregation of the population and to show pattern formation when time increases. We provide for the first time a global existence analysis for an arbitrary number of population species using the entropy method of [15], and we reveal an astonishing relation between the monotonicity of the entropy and the detailed balance condition of an associated Markov chain. The first global existence result is due to Kim [17] who studied the equations in one space dimension, neglected self-diffusion, and assumed equal coefficients (ai j = 1). The existence of global weak solutions in one space dimension assuming a positive definite diffusion matrix was proved in [27], based on Amann’s results. We conjecture that the entropy is bounded for all time for all nonnegative coefficients and nonnegative initial data and that global existence of weak solutions holds for any (positive) coefficients ai j. The choice of the non-diagonal approximation satisfying this inequality is nontrivial, and this construction is our second idea
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