Study On The Quenching Behavior Of Two Nonlinear Parabolic Equations' Solutions

This dissertation is devoted to the study of the quenching phenomenon for the solution of second order parabolic equations. Firstly, we present how this problem was initiated by Kawarada [14] and some application backgrounds. Then we give a survey from following six aspects on the study of this problem during the past 30 years, that is, quenching for the solutions of nonlinear singular parabolic equations, quenching for the solutions of nonlinear parabolic equations with concentrated sources, quenching for the solutions of impulsive parabolic equations, beyond quenching, quenching for the solutions of parabolic equations with time delay, quenching problem in the hyperbolic equations. And then we analyze thoroughly the quenching phenomenon for two types of parabolic initial-boundary value problems.In Chapter 2 we study the quenching behavior for the solutions of a nonlinear parabolic equation with nonlinear outflux, which is an one-dimensional nonlinear parabolic equation with power-law like source term and left boundary value. We concentrate mainly on the possibility of controlling by suitable way the initial data, so as 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 problem must quench at finite time, and the only quenching point is just on the left boundary, provided the initial data satisfy some monotonic conditions.2.We have quenching rate estimation of the type:1u (0, t ) ~ (T ? t ) 2( q+1)(in this paper, we only get the upper estimation of quenching rate). If the initial data are chosen properly, these results indicate that the nonlinear singular source term will not develop singularity when quenching occurs and it indeed has little effect on the change of quenching properties for the solution even if the source may become singular, provided we choose the suitable initial data.In Chapter 3 we consider the quenching phenomenon for nonlinear coupled parabolic systems, with expectation to generalize the results in Pablo [91].We only get some results for the case of non-simultaneous quenching, and for the case of simultaneous quenching, it still need much work .First, we prove that the solution must quench at some finite time, no matter what the initial data would be, and that the derivatives of the solution with respect to the time variable must blow up at quenching time. Then we give the sufficient and necessary conditions for the non-simultaneous quenching. And then for the case of non-simultaneous quenching, we prove the quenching rate estimation v (0, t ) ~ (T ? t) (ifv is the quenching component).Finally, according to the present state of studying parabolic and hyperbolic quenching problem, we raise some critical questions and pave the way for our future study.