Mathematical analysis of thermal runaway for spatially inhomogeneous reactions. (English) Zbl 0543.35047

Lacey, A. A.

SIAM J. Appl. Math. 43, 1350-1366 (1983).

This paper provides the first rigorous proofs of the often conjectured behaviour of the unsteady solutions of the problem (from chemical reactor theory): \[ \nabla^ 2u+\delta f(u)=\partial u/\partial t,\quad x\in D,\quad t>0,\quad \partial u/\partial n+\beta u=0,\quad x\in \partial \Omega,\quad u(x,0)=u_ 0(x),\quad x\in D. \] (Also, the paper provides sharper estimates than proved previously.) Here u is a measure of the dimensionless temperature rise over the ambient, f(u) is the nonlinear dependence of the heat release on the local temperature. It is well known that if \(\int^{\infty}_{b}ds/f(s)<\infty\) for finite b, for example \(f(u)=O(u^{1+\alpha})\) for large u, \(\alpha>0\), then there exists a value of \(\delta\), say \(\delta^*\), for which there is a steady-state solution of the problem, but for \(\delta>\delta^*\) there is not. This paper shows, using the method of upper and lower solutions, that for \(\delta>\delta^*\) the solution of the transient problem ”blows-up” in finite time for all \(u_ 0(x)\), that is, u(x,t)\(\to \infty\) as \(t\to t_ i-0\) and that \(t_ i=O((\delta -\delta^*)^{-{1\over2}})\) as \(\delta \to \delta^*+0\). For values of \(\delta<\delta^*\), lower estimates are obtained for \(u_ 0(x)\) beyond which u(x,t) again exists only for finite values of t. The analysis is carefully and thoroughly carried out and explained well.

Reviewer: G.C.Wake

MSC:

35K55	Nonlinear parabolic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
80A20	Heat and mass transfer, heat flow (MSC2010)
35K20	Initial-boundary value problems for second-order parabolic equations
35B20	Perturbations in context of PDEs

Keywords:

thermal runaway; growth of solutions; unsteady solutions; upper and lower solutions; blows-up