Existence And Properties Of Solutions For Two Fourth-Order Parabolic Equations

This paper mainly studies the existence,uniqueness and long-time behavior of weak solutions for several fourth-order parabolic equations,including the existence of weak solutions for doubly degenerate fourth-order parabolic equations with diffusion coefficients and the global existence and nonexistence of solutions for nonlinear fourth-order parabolic equations with p-Laplacian diffusion term.For the method,the maximum principle and comparison principle do not hold for higher order problems.So the existence of weak solutions is obtained by a series of necessary estimates for the problem 1.For the problem 2,it is necessary to give some growth conditions to the nonlinear term in order to eliminate the influence of the diffusion term on the nonlinear function.Chapter 1 gives the physical background of the problem,the research status,and introduces the main problems and research results in this paper.Chapter 2 studies the existence of solutions for doubly degenerate fourth order parabolic equations with diffusion coefficient.Firstly,the time variable is discretized for the problem with non-degenerate coefficient case and the corresponding semi-discrete fourth-order elliptic problem is solved by constructing extremum element functional.Secondly,the existence and uniqueness of weak solutions for parabolic problems are obtained by energy estimation and two limit processes.Finally,the existence,uniqueness and regularity of the problem are gained by using the regularization method and uniform estimation for the case of degenerate coefficient at the boundary.Chapter 3 studies the existence and long-time behavior of solutions for nonlinear fourth order parabolic equations with p-Laplacian diffusion term.In order to better apply the potential well theory,the following results are given by introducing the corresponding energy functional.When the initial functional is J(u0)<d,I(u0)>0 or J(u0)=d,I(u0)≥0,the global existence and exponential decay of the weak solution are obtained.When J(u0)<d,I(u0)<0 or J(u0)=d,I(u0)<0,the blow-up behavior of the weak solution in finite time is gained.When J(u0)>d,the global existence and blow-up behavior of solutions are given according to different initial conditions.In addition,the uniqueness holds for bounded solutions.The main difficulty is that the p-Laplacian diffusion term in the problem has an essential effect on the nonlinear source term,so the growth condition is added to the nonlinear term.