Blow-up and convergence results for a one-dimensional non-local parabolic problem (Q5938152)

Denote \(l(v):=\int_0^L v(x) dx\) and consider the equation \(u_t-u_{xx}=-a\bigl(l\bigl(u(t,\cdot)\bigr)\bigr)\) on \((0,T)\times(0,L)\) complemented by the boundary conditions \(u(t,0)=0\), \(u_x(t,L)=b\bigl(l\bigl(u(t,\cdot)\bigr)\bigr)\) and the initial condition \(u(0,x)=u_0(x)\). Here, \(u_0\in L^2(0,L)\), \(l(u_0)\geq 0\), and \(a,b\) are locally Lipschitz functions with polynomial growth. The author proves the following two assertions. NEWLINE{\parindent=7mmNEWLINE\begin{itemize}\item[(i)]If \(b\geq 0\), \(b\) is nondecreasing, \(\int^\infty_01/b(s) ds<\infty\), \(\lambda a\leq b\) for some \(\lambda>0\), \(L<L_{\max}(a,b)\), and \(u_0\) is large enough in a suitable sense then the solution blows up in finite time in the \(L^2(0,L)\)-norm. NEWLINE\item[(ii)]If \(a=b\) satisfies some assumptions (guaranteeing, in particular, the existence of two positive equilibria \(u_1<u_2\)) and \(u_1\leq u_0<u_2\) then the solution exists for all \(t>0\) and converges to \(u_1\) in \(H^1(0,L)\) as \(t\to\infty\).NEWLINENEWLINE\end{itemize}}