Waiting Time Phenomena Of Solutions For Degenerate Parabolic Equations

Ever since firstly proposed by Knerr, 1977 [1], the field waiting time phenomena has experienced a rapid development. Under certain assumptions, the interfaces of solutions for degenerate parabolic equations will remain stationary until certain time t^*,which is the so-called "waiting time",and beyond t^* the interfaces will keep moving. It is well-known that for the porous medium equationthe case of m > 1 is called slow diffusion, where the speed of propagation of disturbances is finite, and 0 < m < 1 corresponds to the quick diffusion case,where the speed of propagation of disturbances is infinite. In other words, the the support of solutions run infinitely instantly when 0 < m < 1, and thus the ever studies on the waiting time problem were restricted on the slow diffusion equations.Recently the waiting time problem of various types of degenerate parabolic equations have been studied substantively, and many meaningful results were obtained. Much effort has been invested to get sufficient conditions for the occurrence of waiting time phenomena of solutions for various degenerate parabolic equations, including second order equations and systems in one dimension (allowing for convection term) and multiple dimensions (allowing for absorption term),as well as fourth order equations. As an example, consider the following bounded continuous nonnegative equations,for some u₀.Knerr pointed out that if u₀ is a continuous positive function on (a,b) and is zero outside, then we have if m/(m-1)u₀^m-1≤C(b-x) for some C > 0 and all x near b,t^* > 0;and if m/{m-1)u₀^m-1≥C(b-x)^α,α<2,t^* =0.Later, Juan L.V(?)zquez put forward the sufficient and necessary conditions about this problem in [2], i.e., if u₀ satisfies the following assumptions:(H1) u₀ is a non-negative Borel measure,u₀(?)0;(H2) (?);(H3) u₀ is zero in (0,∞), and ifζ(0) = 0, then t^*>0 if and only ifIn multi-dimensional case, Nicholas D. Alikakos established the sufficient and necessary conditions of the occurrence of waiting time through Harnack-type inequalities, a different approach from the one-space dimension.then t^* > 0 if and only if In [8], Dal Passo, Giacomelli and Griin proposed the use of coneshaped test functions and a simplification of these methods through an extension of the classical Stampacchia lemma [31], which has been shown in [19] to be applicable to the porous-medium equation, higher order doubly degenerate equations of the form:and other thin-film-type equations.In [19], Robert Dal Passo, Lorenzo Giacomelli and G(u|¨)nther Gr(u|¨)n proposed an alternative approach, which can only ensure the properties of waiting time under the assumption of the so-called "exterior cone condition".This approach only requires the validity of the following integral inequality:Herew is a monotone function of u,and w= 0 at u = 0.Not too many results on the estimates of waiting time are available in the literatures. For porous medium equationin one space dimension, and for g(u)-u^m,Aronson, Caffarelli, and Kamin have shown that the waiting time t^* is estimated by provided that the initial data satisfyIn [17], Chipot and Siders have extended the estimate on upper bound to the higher dimensional cases. In particular, their result implies that if the initial support has an " interior cone property " at x₀∈(?)supp(u₀),thenAs in [6], an estimate on the lower bound of the waiting time for the porous medium equation in high dimensions can be naturally derived from the Harnack-type inequalities,In [18], Shishkov and Shchelkov obtained the estimates on the waiting time of solutions for a class of quasilinear degenerate equations in one space dimension. A general approach of determing the quantitative estimation on the lower bound of waiting time is given by Giacomelli and Gr(u|¨)n in [27], which is applicable to a wide class of degenerate evolution operators, including the porous-medium equation,higher order doubly degenerate equations, other thin-film-type equations and also systems, like the drift-diffusion model for semiconductors.In addition, it is also of interest to understand the way in which any part of the boundary first moves and the smoothness of the interfaces.The main goal of this thesis is to discuss and summarize the results discussed before.