On The Quenching Phenomenon For The Solutions Of Nonlinear Parabolic Equations

This dissertation is devoted to the study of the quenching phenomenon for the solu-tions of second order parabolic equations. Firstly we present how this problem wasinitiated by Kawarada[78] and some application backgrounds. Then we give a surveyfrom following 6 aspects on the study of this problem during the past 30 years, that is,quenching for the solutions of nonlinear singular parabolic equations, quenching forthe solutions of nonlinear parabolic equations with concentrated sources, quenchingfor the solutions of impulsive parabolic equations, beyond quenching, quenching forthe solutions of parabolic equations with time delay, quenching problem in the hyper-bolic equations. And then we analyze thoroughly the quenching phenomenon for 3types of parabolic initial-boundary value problems.In Chapter 2 we study the quenching behavior for the solutions of a nonlinearparabolic equation with nonlinear out?ux, which is an one-dimensional semilinearparabolic equation with power-law like source term and left boundary value. We con-centrate mainly on the possibility of controlling by suitable way the initial data, soas to the only occurrence of the singularity of boundary nonlinearity at finite time( i. e. boundary quenching ). The main results are: 1. The solution of the problemmust quench at finite time, and the only quenching point is just on the left boundary,provided the initial data satisfy some monotonicity conditions. 2. We have quench-ing rate estimate of the type: u(0,t)～(T ? t) 2(q1+1), if the initial data are chosenproperly. These results indicate that the nonlinear singular source term will not de-velop singularity when quenching occurs and it indeed has little effect on the changeof quenching properties for the solution even if the source may become singular, pro-vided we choose the suitable initial data.Chapter 3 is concerned with the effect of nonlocal weakly singular absorptionterm on the quenching behavior for the solutions of parabolic equation, where the singular term is the type of logarithm. Firstly, we prove that there exists a criticallength, such that the solution will quench at finite time T, if the coefficient of theproblem under consideration is greater than or equal to this length. Secondly, wederive the quenching rate estimate, by using the Rescaling technique and upper andlower solutions method. Through this estimate we understand that the asymptoticbehavior of the solution near the quenching time is dependent upon its whole values atlevel set {t = T}. Finally, according to the correspondence of the quenching problemand the blow-up problem, we apply our quenching results to a blow-up problem withexponential type of nonlocal term, and get the sufficient condition for the finite blow-up, blow-up point and blow-up rate estimate.In Chapter 4 we consider the quenching phenomenon for a weakly coupledparabolic system with logarithmical weak singularities, with expectation to gener-alize the results of Salin [112] to systems. We only get some results for the case ofnon-simultaneous quenching, and for the case of simultaneous quenching, it still needmuch work. First, we prove that the solution must quench at some finite time, no mat-ter what the initial data would be, and that the derivatives of the solution with respectto the time variable must blow up at quenching time. Then we give the sufficient andnecessary conditions for the non-simultaneous quenching. And then for the case ofnon-simultaneous quenching, we prove the quenching rate estimate v(0,t)～(T ? t)( if v is the quenching component ). Finally, by the transformation and the quenchingresults, we arrive at the blow-up results for the Newmann boundary value problem ofa nonlinear parabolic systems with gradient terms, i. e. : For all the initial data, thesolution will blow up at finite time T, and we have sufficient condition for blow-up tooccur at finite time, and we have blow-up rate estimate: Z(0,t)～? log((T ? t)) (ifZ(x,t) is the blow-up component).Finally, according to the present state of studying parabolic and hyperbolicquenching problem, we raise some critical questions and pave the way for our futurestudy.