Abstract
In this paper, we consider the generalized BBM-Burgers equation with periodic external force in Rn. Existence and uniqueness of time periodic solutions that have the same period as the external force are established in some suitable function space for the space dimension n≥ 3. Moreover, we also discuss the time asymptotic stability of the time periodic solution. The proof is mainly based on the contraction mapping theorem and continuous argument.
Highlights
We investigate existence and asymptotic stability of time periodic solutions to the generalized BBM-Burgers equation with time-dependent periodic external force n n vt − α∆vt − β∆v + γ∆2v + fj(v)xj = ∆g(v) + hj(x, t)xj
We refer to Zhao [16] and [17]
When γ = 0 and hj = 0, Chen and Xue [2] proved that global existence and asymptotic behavior of solutions in one space dimension
Summary
Global existence and optimal decay estimate of solution have been established in [13] They showed that as time tends to infinity, the global solution approaches the nonlinear diffusion wave described by the self-similar solution of the viscous Burgers equation for n = 1. Our main purpose of this paper is to establish existence, uniqueness and asymptotic stability of time periodic solutions to (1.1). Existence and uniqueness of time periodic solutions vper are established by decay properties of solutions operator and the contracting mapping principle, provided that the norm of hj is suitably small. The study of the global existence and asymptotic behavior of solutions to nonlinear evolution equations has a long history and lots of interesting results have been established.
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