Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations
The main goal of this interesting paper is to characterize the occurrence of blow-up and extinction phenomena for the quasilinear heat equation \[ u_t = \bigl( \varphi (u) \bigr)_{xx} \pm f(u) \quad \text{in} \quad \mathbb{R} \times (0,T) \tag{HE}\(_\pm\) \] supplied with the Cauchy initial data \(u(x,0) = u_0(x)\). It is assumed that \(\varphi \in C ([0, \infty)) \cap C^1(0, \infty)\), \(\varphi'(u) > 0\) for \(u > 0\), \(\varphi (0) = 0\), and \(f(u) \geq 0\). For the equation \(\text{(HE}_+)\) with superlinear growth of \(f(u)\) as \(u \to \infty\), a blow-up occurs in finite time: \(\sup_x u(x,t) \to \infty\) as \(t \to T < \infty\). If there exists a nontrivial extension of the solution for \(t > T\), then the blow-up is said to be incomplete. Otherwise, it is called complete. The authors prove that complete blow-up occurs if the function \[ F(u) = u^{- 2} \int^u_1 G(s)ds\quad\bigl( G(s) = \varphi'(s) f(s) \bigr) \] is unbounded as \(u \to \infty\), while the blow-up is incomplete if \(F(u)\) is bounded as \(u \to \infty\) and \(\int^\infty_1 \varphi' (s)s^{- 1} ds < \infty\). The analysis is done in terms of the qualitative behaviour of the family of travelling-wave solutions of the form \(v(x,t) = \theta (\xi)\), \(\xi = x - \lambda t + a\) where the profile function \(\theta\) satisfies a nonlinear ordinary differential equation. As it concerns the Cauchy problem for the equation \(\text{(HE}_-)\) the authors consider the case of singular absorption: \(f(u) \to \infty\) as \(u \to 0\) with a \(C^1\)-function \(f : (0, \infty) \to (0, \infty)\). Then initially positive solutions vanish at some point in a finite time (extinction, quenching). Careful analysis on the same travelling-wave solutions allows the authors to describe precisely occurrence of complete/incomplete extinction for \(\text{(HE}_-)\) in terms of convergence/divergence of the integral \(\int^1_0 G(s)ds\).
scientific article; zbMATH DE number 760198
scientific article