A quasilinear parabolic equation of the reaction-diffusion type with free boundary, M. S. Rasulov 1, and A. K. Norov 1

A quasilinear parabolic equation of the reaction-diffusion type with
free boundary,
M. S. Rasulov 1, * and A. K. Norov 1, **
(Submitted by A. A. Editor-name)
1 Institute of Mathematics named after V.I.Romanovsky, 9 University Street, Olmazor district, Tashkent, 100174, Uzbekistan
Abstract—In this paper, we study the problem for a quasilinear parabolic equation with free boundary. To solve the problem, a priori estimates of the Holder norms are established. On the basic of a priori estimates, the existence and uniqueness of the solution is proved.
2010 Mathematical Subject Classification: 35K10, 35K20, 35K59
Keywords and phrases: Free boundary, Quasilinear parabolic equation, Reaction-diffusion, Asymptotic behavior
1. INTRODUCTION
A large number of physical, biochemical and ecological applications are associated with the reaction-diffusion problem, which is an evolutionary equation in which the spatiotemporal changes in the variable under study are due to diffusion in the spatial variable and nonlinear [15].
The reaction-diffusion model proposed independently by Fischer [9] and Kolmogorov, Petrovski and Piskunov \cite{Aron} is chosen as the basic model of the dynamic theory of population:
(1)
In these works, the authors used an equation to model the spread of a useful genetic trait in a population and proved that there are solutions to a traveling wave, namely solutions of the form , where is the propagation velocity. These results were generalized by Aronson D.G, Weinberger H.F [20] for more general classes of reaction terms.
The free boundary problem for the differential equations has been studied for a long time since the middle of the last century, which describes a class of equations whose domains have a changing surface front, and can be applied to the investigation of the melting of ice in contact with water [], wound healing [] and cancer migration []. Recently, Du and Lin [8] proposed a new approach to investigate whether an invasive species can successfully immigrate in a new environment or not, and studied the following free boundary problem for a reaction-diffusion equation
(2)
where is the free boundary that represents the spreading front of the domain.
Since then there has been extensive works for the reaction-diffusion model of various different types (see. []).
In many studies, the term convection is linear and depends only on the density gradient of the \cite{Gu, RenX} components. However, in general, convection is also affected by the density of the components, which in turn leads to nonlinear convection. In \cite{WangWang}, the authors investigated the free boundary problem for the reaction-diffusion equation with a nonlinear convection term. They obtained the result of a dichotomy and presented a constant asymptotic propagation speed of the expanding front. In [19] has been investigated a quasilinear reaction-diffusion type equation with free boundary.
Motivated by above work, we investigate the quasilinear reaction-diffusion equation with free boundary:
(3)
(4)
(5)
(6)
(7)
where is the free boundary, which is defined together with the function . for any , , , , , , are positive constants. The initial function satisfies , , in and .