Semi-linear evolution equations via positive semigroups
Wolfgang Arendt, Daniel Daners · Discrete and Continuous Dynamical Systems - B · 2024
We study semi-linear evolutionary problems where the linear part is the generator of a positive $ C_0 $-semigroup. The non-linear part is assumed to be quasi-increasing. Given an initial value in between a sub- and a super-solution of the stationary problem, we find a solution of the semi-linear evolutionary problem. Convergence as $ t\to\infty $ is also studied for the solutions. Our results are applied to the logistic equation with diffusion, to a Lotka-Volterra competition model, and to the Fisher equation from population genetics.