The local analytic solution to some nonlinear diffusion-reaction problems GABRIELLA BOGNÁR ERIKA ROZGONYI Department of Analysis Department of Analysis University of Miskolc University of Miskolc 3515 Miskolc-Egyetemváros 3515 Miskolc-Egyetemváros HUNGARY HUNGARY matvbg@uni-miskolc.hu matre@uni-miskolc.hu Abstract: -The positive radially symmetric solutions to the nonlinear problem 0 ) ( 2 = +       ∇ ∇ - u f u u div p in { } , : R x x B n R < ∈ = R , 1 n p < < with δ γ u u u f + = ) ( for 0 ≥ u are considered. We examine the existence of local solutions and give a method for the determination of power series solutions. The comparison of the local analytic and entire solutions is given for some special values of parameters p , n , γ and δ . Key-Words: - Nonlinear partial differential equations, p-Laplacian, non-Newtonian fluid, polytrophic gas, local analytic solutions 1 Introduction In a model for diffusion-reaction problem the concentration of the steady state satisfies , in 0 ) ( n p u f u R ⊂ Ω = + Δ (1) where       ∇ ∇ = Δ - u u div u p p 2 is the p - Laplacian of , u f is an increasing function. The equation above appears in the generalized reaction-diffusion theory [12] and in non- Newtonian fluid theory [10]. In the case of compressible fluid flows in a homogeneous isotropic rigid porous medium the continuity equation is given by ( ) , 0 div = + ∂ ∂ V t  ρ ρ θ (2) where  denotes the density of the fluid, V  the seepage velocity and θ the volumetric moisture content. The linear Darcy’s law is not valid here, since the molecular and ion effects have to take into account in case of a non-Newtonian fluid. Therefore the nonlinear relation P P C V ∇ ∇ - = -1 α ρ  (3) is valid, where P denotes pressure, C and α are positive physical constants. If we consider the fluid as polytrophic gas then we have the relation between the thermal pressure P and the fluid density ρ as γ ρ k P = (4) with positive constant k . Here γ is called polytrophic exponent. We note that for isothermal flows 1 = γ ( 0 = γ corresponds to the isobaric flows). After changing variables and making substitutions in equations (2-4) we obtain . div 2       ∇ ∇ = ∂ ∂ - u u t u p The case 2 > p is called slow diffusion and the case 2 1 < < p , the fast diffusion (see e.g., [20]). When reaction term is added to the diffusion then equation (1) appears in the steady-state case. The asymptotic and numerical solution of problem (1) has been attracted considerable interest in the last decades (see [7], [18], [20]). Nonlinear partial differential equation of type (1) was considered previously for different function . f In paper [4] we considered function WSEAS TRANSACTIONS on MATHEMATICS Gabriella Bognar and Erika Rozgonyi ISSN: 1109-2769 382 Issue 6, Volume 7, June 2008