The global solution of the (2+1)-dimensional long wave–short wave resonance interaction equation

Abstract

Global existence and asymptotic behavior of solutions to a nonlinear wave equation of fourth-orderIn this paper, we prove the existence and uniqueness of the global smooth solution to the ͑2+1͒-dimensional long wave-short wave resonance interaction equation.

Weakly Compact Uniform Attractor for the Nonautonomous Long-Short Wave Equations

“…According to the standard existence theory for the ordinary differential equations, there exists a unique solution of (94). Similar to [4,22], by the a priori estimates in Section 3 we know that { } ∞ =1 converges (weakly star) to a ( , ) which solves (2)∼ (5).…”

Uniform Attractor for the Fractional Nonautonomous Long-Short Wave Equations

“…In this section, we show the unique existence theorem of the solutions. Since uniform a priori estimates have been established in the above section, one can readily get the existence of the solution bÿ's method (see [20,[22][23][24]). We show the theorem and prove it briefly for readers' convenience.…”

“…In this section, we will prove the existence of the strong compact uniform attractor of problem (1)∼(4) applying Ball et al 's idea (see [19,22]). Firstly, we construct a bounded uniformly absorbing set.…”

Uniform Attractor and Approximate Inertial Manifolds for Nonautonomous Long-Short Wave Equations

“…In this section, we show the unique existence theorem of the solutions. Since uniform a priori estimates have been established in the former section, one can readily get the existence of the solution by Galërkin's method (see [9,14,16,19]) or operator semigroup method (see [6]). We show the theorem and prove it briefly for readers' convenience.…”