Blow-Up Of Solutions For Nonlinear Diffusion Equations With Mixed Boundary Conditions

Nonlinear diffusion equations, as an important class of parabolic equations. have profound background.and they are the mathematical formulation of diffusion phenomena appeared widely in nature. They are involved in many mathematical or physical fields of science. such as filtration and dynamics of biological groups and so on. Among these equations, there are two kinds which are most elementary but quite important, i.e. the following Newtonian filtration equations with the typical exampleand the non-Newtonian filtration equations with the typical exampleThe character of both equations is their degeneracy. Comparing to linear equations and quasilinear equations without degeneracy and singularity, such nonlinear diffusion equations with degeneracy reflect even more exactly the physical reality. So. they attractednumerous mathematicians' attention both in China and abroad. They took up with the researches on theory and application of this class of equations, including existence, nonexistence. uniqueness. asymptotic property and Blow-up etc. There have been a lot of corresponding literature dealing with the results. See for exampleThere have been plenty of results in the history of researches on Blow-up property. Now let's have a brief retrospect.Researches about solutions for parabolic equations began with the study of following linear diffusion equation with nonlinear source(I)Early in 1966. Fujia investigated the case f(x.u.t) = up for the C'auchy problem of the above equation[5]. and he proved the Blowup property under certain condition of the exponent p. In 1985. Friedman and McLeod studied the property of the radial solution for equation (1) with Dirichlet zero boundary problem on x [0. T]. where is the bounded area in Rn with 6C2. and f(s) is a positive continuously differential function, and the initial function Y 0. Y G CX(Q). = 0 on . They proved the existence of Blow-up point for two typical cases, i.e. the case f(u) = ?" and the case /) = (u + A)/', where ; 1. A 0(10].Later on. some authors studied the initial-boundary problemBlow-upill one dimension case (2)In 1990, Alberto and Bressan investigated the case f(u) = e" for the Dirichlet zero boundary problem of the above equation on [-1.1] x [0.7") [9]. Compared with [10]. in this paper, the authors pointed out the solution must blow up as long as the initial function is sufficiently large, and the Blow-up time T is dependent on the initial data. Then in 1997. Xoriko Mizoguchi and Eiji Yanagida studied the case f(u) = \up~[u for the Cauchy problem of equation (2). where p > 1[7]. and they proved the Blow-up property under certain conditions of the exponent p.The above researches mostly related to the Dirichlet zero boundary problem, the Cauchy problem or linear diffusion cases, whereas discussions on other types of initial-boundary problems is relatively less. In year of 1993. Mingxin Wang studied the following initial-boundary problem of nonlinear parabolic equation with nonlinear boundary condition[12]. where is a bounded domain in R" with properly smooth, and-?denotes the outward normal derivative. His main result is the vvfollowing:i)if p + q 2. the above problem has a global solution: ii)if j)-f q > 2 and m 1. for a small initial data, the problem has a global solution: but for a large initial data, the solution must blow up in finite time.It is worth while to see that some authors delt with the problem with mixed boundary conditions with the boundary of the domain being divided into two parts. In 1997. by the Green function method. Hu.Bei and Yin. Hong-Ming discussed the Dirichlet-Neumann problem for linear equation with nonlinear boundary condition^ 5]-- = AM. atwhere t > 0. Q is a sector in R2. 9 is apart smooth. - denotesthe outward normal derivative, Fi.F 0. and FI UF2 = O. Theirconclusion isi)if 1 < p < pc. all the positive solutions of the problem must blow up in finite time:ii)if p > pc. for a initial...