Nonlinear Nonlocal Reaction-Diffusion Equations

Abstract

Let $\varOmega \subset \mathbb{R}^{N}$, and J be a nonnegative function defined in Ω ×Ω. We consider the problem

$$\displaystyle\begin{array}{rcl} \left \{\begin{array}{ll} u_{t}(x,t)& =\int _{\varOmega }J(x,y)u(y,t)\mathit{dy} - h(x)u(x,t) + f(x,u(x,t)),\,x \in \varOmega,\;t > 0 \\ u(x,0) & = u_{0}(x),\quad x \in \varOmega,\end{array} \right.& &{}\end{array}$$

(1)

with h ∈ L ^∞(Ω), $u_{0} \in L^{p}(\varOmega )$ and the function f defined as $f:\varOmega \times \mathbb{R}\rightarrow \mathbb{R}$, that maps (x, s) into f(x, s). We assume f globally Lipschitz or f locally Lipschitz in the variable $s \in \mathbb{R}$, uniformly with respect to x ∈ Ω, and f satisfies that there exist $C \in \mathbb{R}$ and D ≥ 0 such that

$$\displaystyle{ f(\cdot,s)s \leq \mathit{Cs}^{2} + D\vert s\vert,\;\forall s \in \mathbb{R}. }$$

The aim is to study the existence and uniqueness and we give some asymptotic estimates of the norm L ^∞(Ω) of the solution u of the problem (1), following the ideas of [2], and we prove the existence of two ordered extremal equilibria, like in [6], which give some information about the set that attracts the dynamics of the solution of (1), for all u ₀ in L ^∞(Ω).