Large Time Behavior Of A Class Of Newtonian Filtration Equations With Sources And Convection

In this paper, we investigate the large time behavior of solutions of nonlinear diffusion equations, which is an important topic in the research of the partial differential equations. The diffusion equations come from a variety of diffusion phenomena appeared widely in nature. They are suggested as mathematical models of physical problems in many fields such as filtration, phase transition, image segmentation, biochemistry and dynamics of biological groups. The research on nonlinear diffusion equations attracts many mathematicians' attention and there have been a lot of results about them.Particularly, people investegated the Newtonian filtration equation and the non-Newtonian filtration equation. The study on the problem with nonlinear sources is an important part in the theoretical study on nonlinear diffusion equations. Compared with the linear equations, the existences of nonlinear terms can heavily affect the properties of the solution, particularly to the long time behavior of the solution. They can cause finite time blow up or extinction to the solution.The focus of this paper is the Fujita exponent theory of a class of Newtonian filtration equations with Sources and Convection. In 1966, Fujita [7] considered the cauchy problem of the following semilinear equationHe proved that: (i) If 1 < p < 1 + 2/N, then no nontrivial nonnegativeglobal solutions exist; (ii) If p> 1 +2/N, then there exist globalpositive solutions if the initial values are sufficiently small.This indicates that the nonlinear exponent p can directly influence the behavior of the solution, the above results are usually called the results of Fujita type and the constant p_c is usually called the critical Fujita exponent.Because the critical Fujita exponent can well describe the long time behavior of the solution, many mathematicians expanded Fu-jita's result to different problems. Besides the cauchy problem on the whole space, people also considered the Dirichlet and the Neumann problem on some special domains, such as the domains with bounded complement and sectorial domains. The results obtained indicated that the Fujita exponent not only depends on the spatial dimension, but also has close relation to the parameters that decribe the shape of the domain. In place of the reaction term u^p, people also considerer other kinds of reaction terms such as t^k|x|^σu^p and cexp(bt)u^p. In addition to the linear diffusion, people also investigated the equations with other diffusion terms, in which the research of the Newtonian filtration equation and the non-Newtonian filtration equation have achieved many results.Many people concerned with the diffusion equations with lower order terms, a typical case of which is the following quasilinear Newtonian filtration equationwith m, q≥1, a(x)∈Rⁿ.In [2] and [22], the author concerned the case when D (?) Rⁿ is a bounded domain. When D is the domain with bounded complement, the large time behavior of the solution will have many differences if m and q are in the different range.In the recent years, there are some progress, especially in the study of the nonlinear equations with convection terms, In [30], the authors investigated the following quasilinear equation with m≥1, p > m, - 1 <λ₁≤λ₂, k∈R, and B₁ is the unit ball in Rⁿ. The equation is subjected to homogeneous Dirichlet or Neumann initial-boundary condition.For the homogeneous Neumann initial-boundary condition, the author proved that the critical Fujita exponent isFor the exterior problem, this was the first time to get the result that the Fujita exponent can be +∞and it's an interesting new phenomenon. For the critical case, the author proved that the critical p = p_c belongs to the blow-up case.The above research is mainly about the slow diffusion, for the fast diffusion, there are few results. In the present paper, we study the following exterior problemwith 0 < m < 1, p > 1,λ≥0, k∈R, and B₁ is the unit ball in Rⁿ. We find that the critical Fujita exponent is determined not only by the spatial dimension and the nonlinearity exponent, but also by the coefficient k of the first order term. In fact, we show that there exist two thresholds denoted by k_∞and k₁ on the coefficient k of the first order term, here When k belongs to (-∞,k_∞), (k_∞,k_∞) and (k₁,+∞), the solution of the problem exhibits completely different asymptotic properties. Actually, we proved thatCompared with the slow diffusion case, this is an interesting new phenomenon and indicates the difference between the fast diffusion and the slow diffusion.In this paper, we mainly used the energy estimate and the supper and sub solution. For the critical case, we use reduction to absurdity.For the integralwe obtain the following estimateUsing this inequality, we get the blow up results.In order to prove the global existence results, we construct the following self-similar supper solution: Using the global existence of this supper solution, we obtain the global existence results with the small initial data.