Anysis For Solutions Of A Degenerate Singular Equations With Initial-boundary Value Conditions

Mainly investigate for this paper is the degenerate singular parabolic equations.This equations is different from equations (system) which we’ve ever seen. It is not onlydegenerate but also singularity. In this paper, we will use some relevant knowledge,methods, and analytical skills which are learned from the textbooks and the relevantliterature to analysis the quality of the solution. And some relevant conclusions areobtained.In the introduction section, we first provide the main problems and related researchbackground which is going to study in this paper. In the meantime, for some theoremsand definition which will be used later in this paper.Detailed introduction is given. It isprepared for later research. And will give a general summary about the chapters.In the second chapter, we have used upper and lower solution to research thequality of global existence of solution to the degenerate singular parabolic equations.Here, we will use the method which is to research the global existence of solution to theelliptical equations. When structure the upper solution, first we converts the parabolicequations which will be researched to elliptic equations. And then research it. At thesame time, for the existence of solution in the overall time, we will estimate the upperbound for the diameterDof the area. And an upper bound for the diameterDisobtained.In the third chapter, we studied the quenching phenomenon of the initial-boundaryvalue problem corresponding to the equations. And the conditions which the solutionshould be satisfied are given when the solution occur quenching phenomenon. And usethe Jensen inequality and the second Green formula estimate quenching time when itssolution occur quenching phenomenon. The upper and lower bound of the quenchingtime are obtained.The main content which is researched in the fourth chapter is estimate for thequenching point set. Due to the equations which are researched in this paper havesingularities. So, we are going to construct a new function, make quenchingphenomenon into blow-up phenomenon. That is we will research the blow-up point setwhich is transformed from phenomenon point set to be researched in this paper. Andfinally we have verified the quenching point set of the solution to the problem which isresearched in this paper is a compact set. Finally, in the fifth chapter, the main content which will be researched is estimatesfor the quenching rate. For we already master many methods and analytical skills forresearch blow-up rates. So, we construct an auxiliary function, when the function occurthe so-called quenching phenomenon, the auxiliary function will be blow-up. So that wecan research the blow-up rate of the auxiliary function first, then reuse the relationshipbetween the original function and the auxiliary function to obtain the quenching rate oforiginal function.