On the \(\omega\)-limit set of a nonlocal differential equation: application of rearrangement theory. (Q331346)

The author investigates the \(\omega\)-limit set of solutions for the nonlinear differential problem NEWLINE\[NEWLINE\begin{cases} u_t=g(u)p(u)-g(u)\frac{\int_{\Omega}g(u)p(u)}{\int_{\Omega}g(u)},\,\,\,x\in\Omega,\,\,t\geq 0,\\ u(x,0)=u_0(x),\,\,\,x\in\Omega,\end{cases}\leqno(1)NEWLINE\]NEWLINE where \(\Omega\subset \mathbb{R}^N\) (\(N\geq 1\)) is an open bounded set, \(g,\,p:\mathbb{R}\to \mathbb{R}\) are continuously differentiable functions and \(u_0\) is a bounded function, which satisfy some additional assumptions. First, he proves the existence and uniqueness of the solution of problem \((1)\), and the uniform boundedness and relative compactness properties in the space \(L^1(\Omega)\). Then he shows that problem \((1)\) possesses Lyapunov functionals and the \(\omega\)-limit set \(\omega(u_0)\) consists of stationary solutions. Under some hypotheses, this \(\omega\)-limit set contains one element, or it contains step functions taking three values.