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Mathematical problems from combustion theory. (English) Zbl 0692.35001

Applied Mathematical Sciences, 83. New York, NY etc.: Springer-Verlag. 177 p. DM 68.00 (1989).
The authors give a very clear and systematic treatment of some basic problems in combustion theory.
In the first chapter the physical and chemical background is given. Then they step forward from the simplest models to more and more involved ones. A special case of the combustion problem leads to the well known Gelfand problem \[ \Delta u+\lambda e^ u=0\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega, \] where \(\Omega\) is some bounded domain of \({\mathbb{R}}^ n\). Existence and uniqueness questions in this and related problems are discussed as well as the geometry of the solution. A next step is then the time dependent problem \(u_ t=\Delta u+\lambda e^ u\) with zero Dirichlet data and vanishing initial distribution. Here the main concern is the blow-up of the solution (when, where, how?). The following chapter then deals with the complete model for solid fuel which is described by a system of two semilinear parabolic equations. After this they discuss the gaseous ignition model which is governed by the equation \[ u_ t-\Delta u=f(u)+\alpha \int_{\Omega}u_ t(x,t)dx,\quad \alpha =const., \] with given initial and boundary values.
The final chapter treats conservation systems for reactive gases.
This book is of interest to a large audience.
Reviewer: R.Sperb

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35K57 Reaction-diffusion equations
35J65 Nonlinear boundary value problems for linear elliptic equations
80A25 Combustion
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B40 Asymptotic behavior of solutions to PDEs
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