On A Class Of Singular Partial Differential Equations With Singular Terms Depending On Gradient

In this paper, we study a class of singular partial differential equations, the singular term is in the form:It’s not difficult to see that this term is singular at points that u=0. Be-cause the singularity depends on gradient (it’s also known as the lower order term), that makes the singular equation is different from the other singular e-quations which singularity depends only on u itself. As we all know, a singular equation can be transformed into a equation without singularity. However, the transformed problem may have some bad properties, for example, the blow-up phenomenon, i.e. the value may become∞. From problems (1-1-5) and (1-1-6) one can see clearly what i mean. The boundary blow-up problems (1-1-5) and (1-1-6) and their elliptic types have been vastly studied in the past few decades and is still attract many scholars’attention. Therefore, research on this class of singular equations is meaningful. For these reasons, in recent four years, this class of singular problems have attract many interest, and there are more and more literatures on it.In this paper, we concern parabolic equations’initial boundary value prob-lem (IBVP)(1-1-1) and elliptic equations’boundary value problem (BVP)(1-1-2). Our research topics are all classic problems, such as existence, uniqueness, stationary problems, long time asymptotic behaviors. This thesis is organized as follows: In Chapter1, we show our main problems of this thesis, review existing results on this class of problems and moreover summarize the main theorems of this thesis.In Chapter2, we mainly study the existence, uniqueness of nonnegative classical solutions of IBVP (1-1-1) and obtain some inequalities. In this chapter we introduce the local analysis method. Combining with regularization method, we proved uniqueness.In Chapter3, firstly we simply proved existence and uniqueness of nonneg-ative solutions of BVP (1-1-2) and get some inequalities by the same method as in the second chapter. Secondly, under some given conditions, we study the long time behavior of the positive classical solution of IBVP (1-1-1), and obtain that when lâ†'+∞it becomes a positive classical solution of BVP (1-1-2).In Chapter4, considering that singular problems can be transformed into boundary blow-up problems, we show readers several questions for further re-search, which are inspired by existing results and unsolved problems that arising from the research of boundary blow-up problems. In the end, we summarize the whole thesis.