Abstract
In this paper, we study properties of solutions to doubly nonlinear reaction-diffusion systems with variable density and source. We demonstrate the possibilities of the self-similar approach to studying the qualitative properties of solutions of such reaction-diffusion systems. We also study the finite speed of propagation (FSP) properties of solutions, an asymptotic behavior of the compactly supported solutions and free boundary asymptotic solutions in quick diffusive and critical cases.
Highlights
Let’s consider properties of the Cauchy problem for the following system of nonlinear reaction-diffusion equa-{ } tions in the domain Q = (t, x) : t > 0, x ∈ RN ( ) = ∂u div ( ) x uk m1−1 ∇u p−2∇u + γ t vβ1, ∂t (1)( ) = ∂v div ( ) x vk m2 −1 ∇v p−2∇v + γ t uβ2,∂ t u (0=, x) u0 ( x) ≥ 0, (2)v (0,= x) v0 ( x) ≥ 0, x ∈ RN, How to cite this paper: Aripov, M. and Sadullaeva, Sh.A. (2015) Qualitative Properties of Solutions of a Doubly Nonlinear Reaction-Diffusion System with a Source
The processes of the reaction-diffusion, heat conductivity, polytrophic filtration of liquids and gas with a source power which is equal to vβ1,uβ2
We study the weak solutions of system (1) which have physical sense: 0 ≤ u, v ∈ C (Q) and x uk m1−1 ∇u p−2∇u, ( ) x vk m2 −1 ∇v p−2∇v ∈ C Q satisfy some integral identity in the sense of distribution [1]
Summary
Let’s consider properties of the Cauchy problem for the following system of nonlinear reaction-diffusion equa-. The process of the reaction-diffusion with double nonlinearity in the case of one equation has been investigated by many authors (see [8]-[15] and the references therein). FSP and blow-up property for equations with variable density ( ) ρ ( x) ∂u = div um−1 ∇u p−2∇u , ( x,t ) ∈ RN +1, ρ ( x)= x −l, l ≥ 0 ∂ t was established in [8] [9]. An asymptotic property of compactly supported solutions (c.s.s.) of the considered problem and the behavior of the free boundary for the case mi + p − 3 > 0, i =1, 2 are obtained. An asymptotic of a self-similar solution for the fast diffusion case (mi + p − 3 < 0, i =1, 2) and a critical case mi + p − 3 =0, i =1, 2 are studied
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