The Long Time Behavior Of The Fractional Long-short Wave Equations

In the paper, the long time behavior of fractional long-short wave equations and(2+1)-dimensional fractional long-short wave equations were studied. We prove the existence of the global smooth solution and uniform attractors for the generalized fractional long-short wave equations with initial and periodic boundary conditions. We prove the existence and uniqueness of the global smooth solution for the(2+1)-dimensional fractional long-short wave equations and the global attractor of these equations.The paper is organized in five chapters.In Part 1, we give the physical background and some basic theories for the fractional calculus, the infinite dimensional dynamic system and the long wave-short wave equations. We recall some known results and briefly describe our work in the present paper.In Part 2, we consider the initial and periodic boundary problem of the generalized fractional long-short wave equations. Firstly, we get uniform priori estimates by using the Gagliardo-Nirenberg inequality, the Young inequality and the Gronwall inequality. Secondly, we prove the existence and uniqueness of the solution for it by using the Gal?rkin method.In Part 3, we prove the existence of uniform attractors for the generalized fractional non-autonomous long-short wave equations. By using the uniform priori estimates and the Gal?rkin method, we prove the existence and uniqueness of the global smooth solution of the solution for the equations. Then we obtain the existence of uniform attractor for it by using the theory of uniform attractor for non-autonomous dynamical systems.In Part 4, we consider the initial and periodic boundary problem of the(2+1)-dimensional fractional long-short wave equations. The uniform priori estimates and the Gal?rkin method are applied to get the existence of the global smooth solution for it.In Part 5, we prove the existence of the global attractors for the(2+1)-dimensional fractional long-short wave equations. Using the theory of the seimgroup and the weak convergence method, we prove the existence of the strongly global attractor for the equations.