Long Time Behavior Of Solutions For Several Evolutional Partial Differential Equations

Many evolutional partial differential equations (PDEs) are used to describe the natural and social phenomena. For example, the Navier-Stokes equation is used to describe turbu-lence and the Black-Scholes stochastic PDE is used to describe the option pricing of the s-tocks. The research on these evolutional PDEs not only can offer us the theoretical foundation for understanding all kinds of natural and social phenomena, but also can give us the effective methods and useful tools to apply and control the natural and social phenomena. This dis-sertation thesis for Ph.D. considers the long time behavior of solutions to several evolutional PDEs, and the main topics contain the existence of global attractor, blow-up phenomenon of solutions, existence of random attractor and so on. The outline is as follows.In Chapter1, we introduce the background and research progresses of several evolutional PDEs, review some concepts and ideas about infinite dimensional dynamical systems and blow-up phenomena of the solutions. Finally we introduce some function spaces and useful inequalities.In Chapter2, we consider a damped BBM equation on the whole real line R. We obtain the existence of global attractor in H1(R)∩W1,P(R) and boundedness of global attractor in H2(R)∩W2,P(R). Since the Sobolev embedding theorems are not compact on R, we use the tailed estimates to prove the solution semigroup of the Cauchy problem is ω-limit compact. Using some harmonic skills such as high-low frequency decomposition and so on, we prove the low frequency part of the solution tends to zero and the high frequency part has higher regularity. Combining with the tailed estimates we obtain the Cauchy problem of the damped BBM equation has global attractor.In Chapter3, we consider the (H2(R), H5(R)) global attractor about a fractional dissi-pative KdV equation on the whole real line R. Compared with the classical KdV equation, the equation has an extra dissipative term, but the dissipation is weaker than the KdV-Burgers equation. There are some difficulties when studying the existence of the global attractor about Cauchy problem of the weak dissipative KdV equation. First, the dissipative term A2au will bring us some obstacles since the fractional power Laplacian is a nonlocal operator. Second, the Solobev embedding theorems are no longer compact on the whole real line R, we should prove the asymptotic compactness of solutions instead of general compact. We establish the tailed estimates to prove the solution semigroup is pre-compact in H2(R), and obtain the existence of (H2(R), H2(R)) global attractor. By the Holder continuity of υt, we prove the the asymptotic compactness of solution semigroup in H5(R). Using the energy conservation law of the KdV equation, smooth effect of solution semigroup and commutator estimates, we obtain an iterative process and prove the solution is bounded in H5(R). Finally, we obtain the existence of (H2(R), H5(R)) global attractor, this conclusion is optimal since the regularity of solution equals the regularity of force term plus three.In Chapter4, we consider the blow-up criterion about the strong solution of the Cauchy problem of the generalized Camassa-Holm equation. Using a blow-up criterion, we obtain a sufficient condition on the initial data such that the strong solution blows up at a finite time, we also give a global existence result of the strong solution. Compared with the literature we overcome two difficulties, one is the nonlinear effects of the higher order nonlinear dissipative term ukuxxx,(k+1)υk-1υxυxx and the convection term (k+2)υkυx, the other is that the sign of the nonlinear term uk-2(t,x) is uncertain, but classical Camassa-Holm equation (k=1) and Novikov equation (k=2) don’t contain this term.In Chapter5, we consider long time behavior about the solutions of a class of stochastic non-autonomous reaction-diffusion equation with additive noise. We establish a criterion about the existence of random attractor. Since the equation is non-autonomous, polynomial growth is arbitrary order, and the solution don’t have higher regularity, we give some new estimates to prove C-condition, then we prove the corresponding dynamical system is ω-limit compact. Finally we prove the existence of random attractor.