Global existence and blow-up of [formula omitted] solutions for a class of nonlinear wave equations with dispersive term

Abstract

In this paper, we study the initial boundary value problem u tt - α u xxt - u xxtt = σ ( u x ) x , ( α ⩾ 0 ) , x ∈ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , u ( 0 , t ) = u ( 1 , t ) = 0 , t ⩾ 0 , where Ω = ( 0 , 1 ) . We first prove that if σ ( s ) ∈ C k ( R ) ( k ⩾ 2 ) and satisfies ( H 1 ) σ ( s ) is bounded below, i.e. σ ′ ( s ) ⩾ C 0 for some constant C 0 ( H 2 ) | σ 1 ( s ) | ⩽ C 1 ∫ 0 s σ 1 ( τ ) d τ + C 2 , where σ 1 ( s ) = σ ( s ) - K 0 s - σ ( 0 ) , K 0 = min { C 0 , 0 } , u 0 ( x ) , u 1 ( x ) ∈ W k , p ( Ω ) ∩ W 0 1 , p ( Ω ) , 1 < p < ∞ , then for any T > 0 the problem admits a unique solution u ( x , t ) ∈ W 2 , ∞ ( 0 , T ; W k , p ( Ω ) ∩ W 0 1 , p ( Ω ) ) . Then the asymptotic behaviors and blow-up of solutions are discussed in detail.