Nonlinear nonlocal reaction-diffusion problem with local reaction
Let \(\Omega\) be a measure metric space. Assume \(u(x, t)\) is the density of some population at the point \(x\in\Omega\) at time \(t\), \(u_0\) is the initial distribution density of the population, \(J(x, y)\) is a positive function defined in \(\Omega\times\Omega\) that represents the fraction of the population jumping from a location \(y\) to location \(x\), per unit time, and \(f: \Omega\times\mathbb{R}\rightarrow\mathbb{R}\) is a nonlinear term which describes the local rate of production/consumption of the magnitude \(u\) at each point \(x\in\Omega\). The aim of this paper is to make a general study of the nonlinear problem \[ \begin{cases} u_{t}(x,t)= \int_{\Omega}J(x,y)u(y,t)~dy+f(x,u(x,t)) \quad\mbox{if }\quad x\in\Omega, t>0, \\ u(x,0)= u_{0}(x) \quad\mbox{if}\quad x\in\Omega. \end{cases} \tag{1} \] Firstly, the authors prove the global existence of solutions for (1) for initial \(u_0\in X\) for some classes of nonlinear terms \(f(x,u)\) and show some similarities and differences between (1) and the following local reaction-diffusion: \[ u_{t}-\Delta u=f(x,u).\tag{2} \] When \(f\) is globally Lipschitz, they prove that solutions of (1) satisfy both weak, strict or strong comparison and maximum principles, depending on the conditions for the kernel \(J\). These results are analogous to (2) and differs when \(f(x, u)\) is locally Lipschitz in the variable \(u\in \mathbb{R}\), uniformly with respect to \(x\in \Omega\), and satisfies a sign condition. Secondly, the authors give some asymptotic estimates of the solution. In particular they prove the existence of two extremal ordered equilibria of (1) in \(L^{\infty}(\Omega)\) which enclose all other equilibria. They also prove that the maximal extremal equilibria is ``stable from above'' and the minimal extremal equilibria is ``stable from below''. Thirdly, for nonnegative solutions they prove that if \(f(x,0)\) is nonnegative then there exists a minimal nonnegative equilibrium such that the solutions of (1) with nonnegative initial data enter asymptotically between these nonnegative extremal equilibria. If moreover \(f(x,0) = 0\) and \(u = 0\) is linearly unstable then every nontrivial nonnegative equilibria is strictly positive. Under some sufficient conditions, the authors prove uniqueness of a positive equilibrium which is globally asymptotically stable for nonnegative solutions. Finally, the authors give some further comments about the asymptotic behaviour of (1) and the lack of asymptotic compactness. In particular, they show that if one has enough compactness to guarantee that, along subsequences of large times, solutions converge \textit{a.e.} \(x \in \Omega\), then it would be possible to prove the solutions approach equilibria and even the existence of a suitable attractor.
scientific article; zbMATH DE number 7481818
scientific article