The Initial Boundary Value Problems For A Class Of Nonlinear Wave Equations With Damping Term

In this paper, we are concerned with the following initial boundary value problemu_tt - 2bu_xxt +αu_xxxx = f(u_x)_x, x∈(0,1), t > 0, (1)u(0, t) = u(1, t) = 0, u_xx(0, t) = u_xx(1, t) = 0, t≥0, (2)u(x,0) =φ(x), u_t(x,0) =ψ(x), x∈[0,1] (3)and the initial boundary value problem of the equation (1)u_x(0, t) = u_x(1, t) = 0, u_xxx(0, t) = u_xxx(1, t) = 0, t≥0, (4)u(x,0) =φ(x), u_t(x,0)=ψ(x), x∈[0,1], (5)whereα> 0, b > 0 are constants,φ(x) andψ(x) are given initial value functions, f(s) is a given nonlinear function, and subscripts x and t indicate the partial derivative with respect to x and t , respectively. Equations of type of (1) are a class of nonlinear wave equations describing the propagation of long waves with the viscosity in the medium with the dispersive effect. It can also be governing the problem of the longitudinal vibration of the 1-D elastic rod.This paper consists of three chapters. The first chapter is the introduction. In the second chapter, we will study the existence and the uniqueness of the global generalized solution and the global classical solution for the initial boundary value problems of (1)—(3) and (1), (4), (5). In the third chapter, we will give the sufficient conditions of blow-up of the solutions for the problem (1)—(3) and the problem(1), (4), (5). The main results are the following:Theorem 1 Suppose that f∈C²(R), and there is a constant C₀ such that f'(s)≥C₀ for any s∈R,φ∈H⁴[0,1],ψ∈H²[0,1], andφ(x),ψ(x) satisfy the boundary conditions(2). Then the problem (1)—(3) has a unique global generalized solutionu∈C([0,T];H⁴[0,1])∩C¹([0,T];H²[0,1])∩C²([0,T];L²[0,1]), (6)where u(x, t) satisfies the boundary value conditions (2) in the generalized sense, and it satisfies the initial value conditions (3) in the classical sense.Theorem 2 Suppose that f∈C⁴(R), f"(0) = 0, and there is a constant C₀ such that f'(s)≥C₀ for any s∈R,φ∈H⁷[0,1],ψ∈H⁵[0,1], andφ(x),ψ(x) satisfy the boundary conditions(2). Then the problem (1)—(3) has a unique global classical solutionu∈C([0,T];C⁴[0,1])∩C¹([0,T];C²[0,1])∩C²([0,T];C[0,1]), (7)where u(x, t) satisfies the initial boundary value conditions (2) and (3) in the classical sense.Theorem 3 Suppose that f∈C²(R), and there is a constant C₀ such that f'(s)≥C₀ for any s∈R,φ∈H⁴[0,1],ψ∈H²[0,1], andφ(x),ψ{x) satisfy the boundary conditions(4). Then the problem (1), (4), (5) has a unique global generalized solutionu∈C([0,T];H⁴[0,1])∩C¹([0,T];H²[0,1])∩C²([0,T];L²[0,1]), (8)where u(x,t) satisfies the boundary value conditions (4) in the generalized sense , and it satisfies the initial value conditions (5) in the classical sense.Theorem 4 Suppose that f∈C⁴(R), and there is a constant C₀ such that f'(s)≥C₀ for any s∈R, fⁱ(0) = 0, (i = 2,4),φ∈H⁷[0,1],ψ∈H⁵[0,1], andφ(x),ψ(x) satisfy the boundary conditions(4). Then the problem (1), (4), (5) has a unique global classical solutionu∈C([0,T];C⁴[0,1])∩C¹([0,T];C²[0,1])∩C²([0,T];C[0,1]), (9)where u(x, t) satisfies the initial boundary value conditions (4) and (5) in the classical sense. Theorem 5 Assume thatK > 2,β> 0, n>1 are constants. (2)φ∈H⁴[0,1],φ∈H²[0,1], andthen the generalized solution or the classical solution u(x,t) of the problem (1)—(3) blows-up in finite time T|^,i.e.,where || ? || denotes the norm of the space L²[0,1].Theorem 6 Let u(x,t) be a generalized solution or a classical solution of the problem (1), (4), (5). Assume that the following conditions are satisfied:(2) f(s)∈C²(R) is an even and convex function satisfyingf(0) = 0, f(A₁) -απ²A₁≥0,(3) Intergralconverges, and B < 1, thenwhere...