A quasilinear diffusive logistic equation with free boundary Rasulov. M. S, Norov. A. Q

A quasilinear diffusive logistic equation with free boundary
Rasulov.M.S, Norov.A.Q
Institute of Mathematics, Tashkent, Uzbekistan;
e-mail: [e-mail]
Systems in the form of reaction-diffusion equations are commonly used in modelling a variety of biological, ecological and epedimic problems. One of the well-known examples is the following diffusive logistic model
(1)
where represents the population density, the coefficient represents the intrinsis growth rate, measures its intraspecific competitia, and is the diffusion rate.
In this article, we study the free boundary problem of a reaction-diffusion equation with nonlinear term:
(2)
where is the advection term, for any , , , , are given positive constants. The initial function satisfies and in is the free boundary to be determined.
In 2010 year Du and Lin first introduced free boundary problem for (1), which is descrabes the expansion of biological populations.
In this article proved some apriore bounds of the solution of (2) by using the theory of nonlinear parabolic equation, then the global existence and uniqueness are obtained. And we prove the spreading-vanishing dichotomy and give its criterion by the method of eigenvalue problems and constructing the upper and lower solution.
Theorem. Let be the solution of the free boundary problem (2). Then following holds.

1. 
   Vanishing: and
   
2. 
   Spreading : and uniformly for in any bounded set of where is the unique positive solution of the following problem
   

R E F E R E N C E S
1. Du Y.H, Lin, Z.G. Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 2010, Vol. 42, pp. 377-405.