Asymptotic behaviour of a population model with local, concave growth on Lipschitz domains

We examine the asymptotic behaviour of u˙=dAu+f(u) for positive initial values u(0)>0u(0)>0. Here A   is the generator of an exponentially bounded semigroup on L∞(Ω)L∞(Ω) with several additional properties. A particular example is the Laplace operator ΔΩR,∞ with Robin boundary conditions on a Lipschitz domain defined on L∞(Ω)L∞(Ω). Moreover, f:L∞(Ω)→L∞(Ω)f:L∞(Ω)→L∞(Ω) is a local, continuously Fréchet-differentiable, and strictly concave map with f(0)=0f(0)=0. We will analyse the asymptotic behaviour of the solutions as t→∞t→∞ and its dependency on the diffusion coefficient d>0d>0.