Abstract
Abstract In this paper, we investigate the dynamical behavior of the initial boundary value problem for a class of generalized hyperelastic-rod equations. Under certain conditions, the existence of a global solution in H 3 is proved by using some prior estimates and the Galerkin method. Moreover, the existence of an absorbing set and a global attractor in H 2 is obtained.
Highlights
Camassa and Holm [ ] first proposed a completely integrable dispersive shallow water equation as follows: ut – uxxt + uux + kux = uxuxx + uuxxx. ( . )The C-H equation ( . ) was obtained by using an asymptotic expansion directly in the Hamiltonian for the Euler equations in the shallow water regime and possessed a biHamiltonian structure and an infinite number of conservation laws in involution
The three coefficients σ, σ, and σ are constants determined by the pre-stress and the material parameters, If τ
Motivated by the references cited above, the goal of the present paper is to investigate the initial boundary problem of the following equation: ut – uxxt + [G(u)](x ) = uxuxx + uuxxx, t >, x ∈, ( . )
Summary
Holden and Raynaud [ ] studied the following generalized hyperelastic-rod equation: ut – uxxt + f (u)x – f (u)xxx +. Motivated by the references cited above, the goal of the present paper is to investigate the initial boundary problem of the following equation: ut – uxxt + [G(u)](x ) = uxuxx + uuxxx, t > , x ∈ , u( , x) = u (x), x ∈ , where = [ , L]. Φm}, Pm is the orthogonal projection from H to Hm. Through the Galerkin method, we can obtain the following ordinary differential equations by
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