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

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
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