Quenching behavior for degenerate parabolic problems (Q2724364)

Let \(T \leq \infty\), \(a > 0\), \(\Omega = (0,a) \times (0,T)\), \(\partial\Omega\), be the parabolic boundary \(([0,a]\times \{0\})\cup (\{0,a\}\times (0,T))\) of \(\Omega\), and NEWLINE\[NEWLINE Lu \equiv u_{xx} - x^qu_t NEWLINE\]NEWLINE where \(q\) is any real number. It is considered the following initial-boundary problem NEWLINE\[NEWLINE \begin{gathered} Lu = - f(u)\quad \text{in } \Omega,\\ u(0,t) = 0 = u(a,t) \quad \text{for } t > 0,\\ u(x,0) = u_0(x) \quad \text{for } x \in (0,a), \end{gathered} NEWLINE\]NEWLINE where \(u_0\) is the initial data with \(u_0(0) = 0 = u_0(a)\), \(0\leq u_0\leq c_0 < c\) and \(u_0'' + f(u_0) \geq 0\) while \(f\) satisfies these properties: \(f(0) > 0\), \(f' > 0\), \(\lim_{u\to c -0}f(u) = \infty\) and \(\int_0^c f(u)du = M < \infty\). The author proves that the set \(S_q\) of quenching points is bounded away from the lines \(x = 0\) and \(x = a\), and \(S_q\) lies in a compact subset of \((0,a)\) depending on \(f\), \(a\) and \(u_0\). It is shown that with certain forms of initial data \(u_0\) (including \(u_0\equiv 0\)), the problem can have single-point quenching only. The question on an upper bound for the quenching time is also discussed.