Quenching problem of a functional parabolic equation (Q2724366)

The author studies a quenching problem for a parabolic boundary value problem with functional NEWLINE\[NEWLINE\begin{cases} u_t-Lu= f(u,Ku)\quad\text{in}\quad(0,\infty) \times \Omega_\alpha,\\ u(t,x)=0\quad \text{on}\quad (0,\infty) \times \partial \Omega_\alpha,\quad u(0,x)= u_0(x)\geq 0\quad\text{in}\quad\Omega_\alpha. \end{cases}\tag{1}NEWLINE\]NEWLINE Here \(L\) is a self-adjoint elliptic operator \(Lu=\sum_{i,j} \partial_j(a_{ij} (x)\partial_iu)\) and \(\{\Omega_\alpha: \alpha>0\}\) is a family of smooth bounded domains in \(\mathbb{R}^n\), where \(\bigcup_\alpha \Omega_\alpha= \mathbb{R}^n\), \(\text{diam} (\Omega_\alpha)\to 0\) as \(\alpha\to 0\) and \(\overline \Omega_\alpha \subset \Omega_\beta\) for \(\alpha <\beta\). Further \((Ku)(x)= \int_{\Omega_\alpha} k(x)u (t,x)dx\) with continuous \(k(x)\geq 0\) and \(\int_{\mathbb{R}^n} k(x)dx\leq 1\), and \(f(u, v) \in C^1\) in \([0,c)^2\), \(f\) increasing in \(v\), \(f(0,0)>0\) and \(f(u,v)\to \infty\) as \(u\to c\) for all \(v\). Here \(c>0\) is a fixed constant, and quenching on \(\Omega_\alpha\) of the solution \(u\) of (1) means that \(\max_xu(t,x)\to c\) as \(t\to T\), where \(T\in (0,\infty)\). An example is \(f(u,v)=(1+ av^\beta)/(1-bu^\gamma)\) with \(a,b,\beta, \gamma>0\). If in the steady-state problem NEWLINE\[NEWLINE-Lu= f(u,Ku)\quad\text{in}\quad\Omega_\alpha, \quad u(x)=0\quad\text{on}quad\partial \Omega_\alpha \tag{2}NEWLINE\]NEWLINE there is a pair of upper/lower solutions \(0\leq \widetilde u\leq \widehat u\) for (2), then there is a minimum solution \(\underline u\) and a maximum solution \(\overline u\) of (1) between \(\widehat u\) and \(\widetilde u\). NEWLINENEWLINENEWLINELet \(\alpha^*\) be the supremum of all \(\alpha>0\) such that (2) has a positive solution. Then for any \(\alpha< \alpha^*\) there is a positive minimum solution \(\underline u(x;\alpha)\) and maximum solution \(\overline u(x; \alpha) < c\). If in addition \(f_u(u,v)\geq\varepsilon>0\) in \((0,c)^2\), then \(\alpha^* < \infty\) and for \(\alpha> \alpha^*\) there is no positive solution to (2). For the quenching problem (1) the main theorem states that for \(\alpha< \alpha^*\) and \(u_0(x) \leq\underline u(x)\) a unique global solution exists, and it converges to \(\underline u(x)\) as \(t\to\infty\); in case \(\alpha>\alpha^*\), for any \(u_0 (x)<c\) the solution \(u(t,x)\) of (1) quenches.