On solutions of a singular diffusion equation (Q1567534)

Consider the initial-boundary value problem \(u_t=\nabla(a(u)\nabla u)\) in \(\Omega\times(0,T)\), \(\partial u/\partial\nu=g(x,t)\) on \(\partial\Omega\times(0,T)\), \(u(x,0)=u_0(x)\) in \(\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb R^n\), \(g<0\), \(u_0>0\) and \(a:(0,\infty)\to\mathbb R\) fulfils \(a>0\), \(a'<0\), \(a''\geq 0\), \(a(u)\to+\infty\) as \(u\to 0+\). The solution of this problem is said to quench if it tends to zero and one of its derivatives blows up. The authors show that under some regularity and compatibility assumptions on \(g\) and \(u_0\), the problem does not possess global solutions. If, in addition, \(g_t\leq 0\) and \(u_t\leq 0\) then the solution quenches in a finite time \(T\). If \(\Omega\) is a ball, \(u\) is radial and \(u_r,u_t\leq 0\) then quenching occurs only on the boundary \(\partial\Omega\), and for \(a(u)=u^{-\beta}\), \(0<\beta\), the quenching rate is \((T-t)^{1/(\beta+2)}\). The problem is studied by using the transformation \(v=\int_u^M a(s) ds\), where \(M\) is an upper bound for \(u_0\).