Blow-up And Initial Trace Of Solutions To Several Classes Of Nonlinear Parabolic Equations

Nonlinear parabolic equations can be used to describe a lot of diffusion phenomenathat occur in nature, and have attracted the attentions of many researchers. This thesis isdevoted to the study of several classes of nonlinear parabolic equations. From thesecond chapter to the fifth chapter, we discuss the blow-up properties of solutions tosome kinds of nonlocal parabolic equations (systems), including the conditions forglobal existence or nonexistence of solutions, blow-up rate, uniform blow-up profile,blow-up set and the lower bound of the blow-up time. In the sixth chapter, we discussthe local existence, local regularity and initial trace for a doubly degenerate parabolicequation with initial data measures.In Chapter1, we introduce the background and the development of the relatedtopics and summarize the main content of the present thesis.In Chapter2, we study a one-dimensional doubly degenerate parabolic equationinvolving weighted nonlocal boundary condition. The main purpose of this chapter is todeal with the influences of the weight functions in the nonlocal boundary condition tothe blow-up properties of the solutions. We first give the sufficient conditions forblow-up in finite time by constructing various supersolutions and subsolutions. Then, byintroducing a masterly transformation, we transfer the effect of the nonlinear diffusionand obtain the precise blow-up rate estimate. In addition, based on some classical ideasof Souplet, we get the uniform blow-up profile near the blow-up time and prove thetotal blow-up phenomenon.(The main results of this chapter are published in Appl.Anal.,2011,90:1373-1389)In Chapter3, we concern with a class of semilinear parabolic equation withnonlinear memory and nonlinear nonlocal boundary condition. Firstly, we give a newand suitable comparison principle, and establish the local existence in time anduniqueness of the classical solution by using Green’s function and the contractionmapping principle. Secondly, we deeply discuss the effects of the weight function andnonlinear index in the nonlocal boundary condition on the global existence andnonexistence of solutions, and analyse the substaintial difference between nonlinearnonlocal boundary and linear nonlocal boundary. Finally, we obtain the blow-up rate.(The main results of this chapter are published in Electron. J. Qual. Theory Differ. Equ.,2010,51:1-17) In Chapter4, we first consider a pseudoparabolic equation. By constructing asuitable energy function and using an improved convex method, we prove that theenergy function will tend to infinite provided that the initial datum is sufficiently largein some sense. Then, we investigate the blow-up critical exponent for a p-Laplacesystem with nonlocal reaction terms and inner absorptions. By evaluating the effects ofthe various nonlinearities, initial data and the geometric property of the domain on theblow-up behavior, we obtain a critical exponent.(The results of the first part of thischapter are submitted to Appl. Math. Lett., and the results of the second part arepublished in Bound. Value Probl.,2011,29:1-14)In Chapter5, we focus on the lower bound of blow-up time for a nonlinearnonlocal porous medium equation. By using the differential inequality technique, we getthe optimal estimate of the lower bound of the blow-up time.(The main results of thischapter are published in Acta Math. Sci.,2012,32B:1206-1212)In Chapter6, we study a class of degenerate parabolic equation with initial datameasure, and obtain the local existence, the local regularity of the solution and initialtrace for the supercritical case.(The main results of this chapter are preprinted)...