The Existence And The Non-existence Of Global Solutions Of A Free Boundary Problem

We study a free boundary problem of parabolic equations with a positive parameter r included in the coefficient of the derivative with respect to the time variable t(see (1.2)-(1.7) in section 1), This problem arises from some reaction-diffusion system.Many mathematicians, D.Hilhorst, Y.Nishiura, M.MimuraM, investigate the well-posedness of the solution of this equation. Basing on it, we try to discuss the distribution of this solution. When r is a proper constant and v0 a proper function,the global or local solutions can be obtained respectively with variance of 0.It is inspired to think about the question that how about the solutions when r,an importent parament in the reaction-diffusion equation,is chosen to be different values.In this thesis,we can get the equivalent form of ' by the concrete form of solution of (1.2)-(1.7) and uniform estimates.Then the main idea(theorem1.1 and 1.2) is obtained.Our main results are the following: if is large enough, the solutionexists for 0 < t < +00,that is global solution; while, if T is small enough, the solution exists only in finite time,that is local solution.