The Global Attractors Of Two Nonlinear Equations

This thesis is concerned with the existence and uniqueness of solution, and global attractor's existence of two nonlinear parabolic equations in R 2 and R n. The studies of attractors in whole spaces started with article [1] written by A.V.Babin and M.I.Vishik in 1990, where they studied the equation (?)_tu = γΔu-f(u)-g-λu, by introducing weighted spaces which are most important there , because every result was given in these spaces. From then on, a lot of people have come into this fields, e.g. Bixiang Wang [2] in 1999 imposed some requirements on the nonlinearity which ensured the norm of the solution outside of bounded domains was arbitrarily small uniformly for large time, which, together with the compactness imbedding in bounded domains and J.Ball's method, yielded the existence of attractors in whole spaces. Ferderic Abergel[3] in 1990 obtained the existence of attractors for u_t-υΔu+αu+u(l·▽u)=f. He required the outer forced term satisfy some decay property and so the norm of the solution outside of bounded domains was arbitrarily small uniformly for large time. The first result of this thesis is obtained by borrowing some ideas in Ferderic Abergel. We avoid the requirement of decay of the outer force term by combining the method of J.Ball and the decomposition of the solution. The second result is concerned with an equation with a general nonlinear term. The equations we discuss are: u_t -γΔu + αu + u (l·▽u ) = f ( x , t) u (. , 0)= u₀ (1) u_t-γΔu + αu + f (u , â–½u )= g u (. , 0)= u₀ (2)...