Dissipative property for non local evolution equations (Q2418794)

The authors consider the nonlocal evolution equation \[ d_tu(x,t)=-u(x,t)+g(\beta K (f(u))(x,t)+\beta h), \] where \(x\in\Omega\), \(t\in [0,\infty )\), \(u(x,t)=0\) for \(x\) outside \(\Omega\) and all \(t\), \(u(x,0)=u_0(x)\) for \(x\in \mathbb{R}^n\). Here \(u\) is a real function on \(\mathbb{R}^n \times t\in [0,\infty)\), \(\Omega\) a bounded smooth domain, \(h,\beta\) are non-negative constants, \(f,g: \mathbb{R} \to \mathbb{R}\) are locally Lipschitz and \(K\) an integral operator with symmetric non-negative kernel given by \[ Kv(x)=\int_ {\mathbb{R}^n}J(x,y)v(y) dy, \] where \(J\) is symmetric, nonnegative of class \(C^1\) with \(\int_{\mathbb{R}^n} J(x,y) dy=\int_{\mathbb{R}^n} J(x,y)dx =1\). Assuming growth conditions on \(f,g\) the authors prove that the equation is well-posed. It is also shown that the equation generates a \(C^1\) flow in a space \(X\) isometric to \(L^p(\Omega)\). The existence of a global attractor is obtained. This is proved using a result by \textit{R. Temam} [Infinite-dimensional dynamical systems in mechanics and physics. New York etc.: Springer-Verlag (1988; Zbl 0662.35001)]. Certain comparison and boundedness results are obtained. The proofs are lengthy but straightforward.