| In this thesis,we study the properties of solutions for some nonlinear degenerate parabolic equations,including the existence,regularity and uniqueness of weak solutions.The full text is organized as follows:In the first chapter,we summarize the research background and overseas and domestic research status,state the main results of this thesis.In the second chapter,we study the following nonlinear parabolic equations with nonstandard growth conditions:Problem（P1）is closely related to the model of electror-heological fluids model,with variable exponents of nonlinearity.Thus it is natural to solve problem（P1）under the framework of Sobolev spaces with variable exponents.In this chapter,we will use the Galerkin approximation method to prove the existence and uniqueness of bounded solutions to problem（P1）,which generalizes the corresponding results in the constant exponents.In the third chapter,we mainly study the following nonlinear degenerate parabolic equations:It is easy to see that the main operator is not coercive and degenerate as the solution u tends to infinity,which means that the classical Leray-Lions theory can not be used to prove the existence of solutions.To overcome this difficulty,we will first consider a class of non degenerate and forced approximation problem at infinity.We obtain the Lr（or L∞）estimates for the solutions,finally the compactness theory will be used to prove the existence and regularity of weak solutions（or bounded weak solutions）to poblems.In the fourth chapter,we study the following doubly degenerate parabolic equations:Obviously the main operator is degenerate for{（x,t）∈ QT:u（x,t）= 0},and it may be not coercive where u tends to infinite.To overcome this difficulty,we will prove the existence results to（P3）by using approximations twice,we shall first prove the existence results for the case divg ∈L1（QT）,then we shall prove the existence results to（P3）for the case g ∈{Lp’q（QT）)N.To prove the uniqueness result,we will use the doubling variable method. |
