## Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition.(English)Zbl 0699.35151

The authors study a parabolic initial-boundary-value problem modeling the evolution of the temperature u in an exothermic chemical reaction. The reaction term is given by $$\lambda$$ f(u), under some specific assumptions of f(u), where $$\lambda$$ is a positive parameter.
The authors are especially interested in finding under what circumstances “complete blow-up” may occur. In particular they obtain new conditions which are sufficient for complete blow-up without the requirement $$f(0)=0$$. One of their main conclusions is that even if the domain $$\Omega$$ and f are such that partial blow-up cannot be ruled out it will almost never actually occur. Physically they conclude that if thermal runaway occurs at some point then ignition of a thermal explosion almost certainly takes place immediately.
Reviewer: C.Schmidt-Lainé

### MSC:

 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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