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Existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium. (English) Zbl 0971.76077

The authors consider a model which describes heat explosion in a non-moving heterogeneous medium, consisting of reacting particles surrounded by a gas. This model is represented in the stationary case by the following elliptic-algebraic system of equations \(F(u_1)-\alpha(u_1-u_2)=0\), \(\Delta u_2-\alpha(u_1-u_2)=0\), which can be understood as equilibrium of heat production and heat loss in each phase. Here \(u_1\) and \(u_2\) are the dimensionless temperatures of particles and gas respectively, the parameter \(\alpha\) is the rate of heat exchange between two phases, and the nonlinear function \(F(u_1)\) characterizes the rate of heat production in the particles’ phase. These equations are considered in a bounded domain \(\Omega\subset\mathbb{R}^m\) with a sufficiently smooth boundary \(\partial \Omega\), and the boundary condition is \(u_2\mid_{\partial\Omega}=0\). The function \(F(u_1)\) is supposed to be positive for \(u_1\geq 0\), the parameter \(\alpha\) is also positive.
The basic question in the theory of heat explosion is to find critical conditions for existence of solutions depending on parameters of the problem and, in particular, on the size of the domain. Here the authors suppose that the domain \(\Omega\) is star-shaped. Three types of solutions are found: classical, critical and multivalued. Regularity of solutions is studied as well as behaviour depending on the size of the domain and on the coefficient of heat exchange between two phases. Critical conditions for existence of solutions are found for arbitrary positive source function.

MSC:

76N15 Gas dynamics (general theory)
80A20 Heat and mass transfer, heat flow (MSC2010)
35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35K57 Reaction-diffusion equations
35Q80 Applications of PDE in areas other than physics (MSC2000)
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References:

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