Abstract
In this paper we consider the Cauchy problem for the stochastic modified Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in H^{s}(mathbb{R}), sgeq -1/4. Moreover, we get the global existence for L^{2}( mathbb{R}) solutions. Due to the non-zero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.
Highlights
In this paper, we investigate the Cauchy problem for the stochastic modified Kawahara equation: ∂u ∂t + α ∂5u ∂ x5 β ∂3u ∂ x3 γ ∂u ∂x μu2 = Φ
6 Summary and discussion This paper is devoted to employing the Fourier restriction method, the Banach contraction principle and some basic inequalities for investigating nonlinear stochastic partial differential equations (SPDEs) and for proving local and global well-posedness results for their solutions in convenient function spaces
Our attention is focused to the stochastic modified Kawahara equation (1), which is a fifthorder shallow water wave equation considered in a random environment
Summary
We investigate the Cauchy problem for the stochastic modified Kawahara equation:. This paper focuses on the case when the forcing term is of additive white noise type This leads us to study the stochastic fifth-order shallow water wave equation (1). The first well-posedness result for the Kaup–Kupershmidt equations was presented by Tao and Cui [19] They proved that their Cauchy problems are locally well-posed in Hs(R) for s > 5/4 and s > 301/108, respectively. De Bouard and Debussche [22] considered the stochastic KdV equation forced by a random term of white noise type They proved existence and uniqueness of solutions in H1(R) and existence of martingales solutions in L2(R) in the case of additive and multiplicative noise, respectively. The goal of this paper is to investigate the Cauchy problem of the stochastic modified Kawahara equation (1), where the random force is of additive white noise type.
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