Localization and blow up of thermal waves in nonlinear heat conduction with peaking (Q1092329)

We consider the initial-boundary value problem: \[ u_ t=(u^ m)_{xx}\quad for\quad (x,t)\in (0,\infty)\times (0,t), \] \[ u(x,0)=u_ 0(x)\quad for\quad x\in [0,\infty),\quad u(0,t)=\psi (t)\quad for\quad t\in [0,T), \] where \(m>1\), \(0<T<\infty\), and \(u_ 0\) and \(\psi\) are nonnegative continuous functions satisfying the compatibility condition \(u_ 0(0)=\psi (0)\) and \(\psi\) (t)\(\uparrow \infty\) as \(t\uparrow T\). If \(u_ 0\) is bounded, it is known that this problem has a unique nonnegative generalized solution, u(x,t). Furthermore, if \(u_ 0\) has compact support, the solution possesses a free boundary \(\zeta (t)=\sup \{x\in [0,\infty):\) \(u(x,t)>0\}.\) In the context of nonlinear heat conduction, a solution u(x,t) with such a free boundary is often referred to as a thermal wave. We say that a thermal wave is localized if \(\lim _{t\uparrow T}\sup \zeta (t)<\infty.\) Under the assumption that \(\psi\) is monotonic increasing, we show that localization occurs if and only if \(\lim _{t\uparrow T}\sup [\int ^{t}_{0}\Psi ^ m(s)ds/\Psi (t)]<\infty.\) We also discuss the nature of the blow-up set \(\Omega =\{x\in [0,\infty):\) \(\lim _{t\uparrow T}\sup u(x,t)=\infty \}.\) We show that if localization does not occur, then \(\Omega =[0,\infty)\). On the other hand, if localization does occur, then \(\Omega\) is a bounded interval. In this case, estimates of the size of \(\Omega\) are obtained. These indicate the circumstances under which \(\Omega\) may or may not reduce to the single point \(x=0\).