Global Existence And Blow-up Of Solutions To A Porous Medium System With Nonlocal Boundary Condition

As an important class of partial diffusion equations, nonlinear diffusion equations come from a variety of diffusion phenomena appeared widely in nature. They arise naturally from many fields such as chemistry, physics and dynamics of biological groups. In the past few decades, the study in this direction have attracted a large number of mathematicians both in China and aboard, and remarkable progress has been achieved on the local existence of classical solutions, global existence, blow-up as well as estimates on blow-up rate and blow-up time, which enrich enormously the modern theory of partial differential equations. Until now, the study of reaction-diffusion equations is still a very active research area.There have been a lot of works dealing with properties of classical solutions to partial differential equations or systems with local boundary conditions over the past few years. However, there are also some important phenomena modeled by parabolic equations or sys-tems which are coupled with nonlocal boundary conditions in mathematical modelling such as thermoelasticity theory. In this case, the solution u(x, t) could be used to describe entropy per volume of material.Early in 1986, A. Friedman studied the following type of single equation with nonlocal boundary condition He established the global existence of solution and also discussed its monotonic decay prop-erty when g(x, u)=c(x)u with c(x)≤0 and∫Ω|k(x,y)|dy<1 for all x∈(?)Ω. Later in 1992, Deng in his work established the comparison principle and proved local existence of classi-cal solution to Problem (1) with general reaction term g(x, u). Furthermore, he removed the assumptions c(x)≤0 and∫Ω|k(x, y)|dy<1 for the case of g{x, u)=c(x)u, and proved that the solution exists globally and may increase at most exponentially with t. In 2000, Seo gave a global upper bound of solutions to Problem (1) in the case of g(x,u)= cu (where c is a constant without any restriction) and k(x,y)≥0. In the case of c≥0 and‖k(x,·)‖1> 1, he also investigated the decreasing property of boundary values. When g(x, u) is superlinear in u, the solution to Problem (1) might blow up in a finite time. Seo investigated Problem (1) in the special case g(x, u)= g(u), and gave some sufficient conditions for the positive solutions to blow up in finite time, by using supersolution and subsolution method. The blow-up rate estimates for g(u)= up and g(u)= eu were also derived.There are also some other works on equations or systems with nonlocal boundary con-ditions. For instance, Pao studied the asymptotic behavior of solutions to a class of diffusion equations with nonlocal boundary condition in 1998, and later gave the numerical solutions.The studies mentioned above show that the increasing or decay properties of solution to Problem (1) depend mainly on the growth of g(x, u) with respect to u, which is quite similar to general semilinear equations with homogeneous boundary conditions. On the other hand, however, due to the appearance of the nonlocal boundary conditions, the asymptotic behavior of solutions depend heavily on the weight function k(x, y) as well.In 2007, Wang et al. studied the following porous medium equation with nonlocal boundary condition and local reaction term where m, p> 1 are constants and u0(x) and k(x,y) satisfy the same assumptions as given in Problem (1.1). They proved that if∫Ωk(x,y)dy≥1 for x∈(?)Ω, the solution to Problem (2) blows up in finite time, while if∫Ωk(x, y)dy<1, there may exist both global solutions and blow-up solutions to Problem (2).The main purpose of this paper is to study the global existence and blow-up properties of the following problem where m, n> 1, a, b, p, q> 0 are constants andΩis a bounded domain in RN (N≥1), with smooth boundary (?)Ω. k1(x,y), k2(x,y) (?) 0 are nonnegative continuous functions defined for x∈(?)Ωand y∈Ω, while u0(x), v0(x) are positive continuous functions and satisfy the compatibility conditions u0(x)=∫Ωk1(x,y)u0(y)dy and v0(x)=∫Ωk2(x,y)v0(y)dy for x∈(?)Ω.We say that a vector valued function,(u,v) is a classical solution of Problem (3) if (u, v)∈[C2,1(Ω×(0, T))∩C(Ω×[0, T))]2 for some T:0 1.Theorem 2 Assume that∫ok1(x,y)dy,∫Ωk2(x,y)dy>1 for x∈(?)Ω. Then (u,v) blows up in finite time provided that pq> 1.Theorem 3 Assume that∫Ωk1(x,y)dy,∫Ωk2(x,y)dy<1 for x∈(?)Ω.(ⅰ) If pqmn, then the solution to (3) exists globally provided that (u0, v0) or a, b are small, while it blows up in finite time if (u0, v0) is large enough.In Section 4, we assume that the initial datum (u0, v0) satisfies anther assumption (H) as follows:(H) u0, v0∈C2+a(Ω) for some 0<α< 1 and there exists a constantδ>δ0> 0 such that for some constantsδ0, k1, k2. The main result in this section is the following theorem on the blow-up rate. Theorem 4 (Blow-up rate estimates.) Suppose that (u0, v0) satisfies the assumption (H), p, q> 1 and∫Ωk1(x, y)dy≤1 (i=1,2). If (u, v) is the classical solution to (3) and blows up in a finite time T, then there exist positive constants C1-C4, such that...