The Initial Value Problem For A Class Of Nonlinear Wave Equation Of Fourth-Order

In this paper, we are concerned with the following initial value problems: v_tt－a₁v_xx－a₂v_xxt－a₃v_xxtt = f(v_x)_x, x∈R, t > 0, (1) v(x,0) = v₀(x), v_t(x,0) = v₁(x), x∈R (2) and v_tt－c₁v_xx－c₂v_xxtt = f(v_x)_x, x∈R, t>0, (3) v(x, 0) = v₀(x), v_t(x,0) = v₁(x), x∈R, (2) where a₁, a₂,a₃, c₁, c₂ > 0 are constants, f(s) is a given nonlinear function, V₀(x) and v₁(x) are given initial value functions,and subscripts x and t indicate the partial derivative with respect to x and t, respectively. Equation (1) is a nonlinear evolution equation. There exists the equation (1) in the study of nonlinear waves in the elastic rods .In the study of strain solitary waves in nonlinear elastic rods ,there exists the equation (3). We make a change of variables x = (a₃)^1/2y, t = t, (4) then the initial value problem (1), (2) becomes u_tt－αu_xx－βu_xxt－u_xxtt =φ(u_x)_x, (5) u(x,0) = u₀(x), u_t(x,0) = u₁(x). (6) Let x =(c₂)^1/2y, t = t (7) then the initial value problem (3), (2) becomes u_tt－δu_xx－u_xxtt = g(u_x)_x, (8) u(x,0) = u₀(x), u_t(x,0) = u₁(x). (6)In this paper,we only study the existence and uniqueness of the global generalized solution , the global classical solution and blow-up of the solution for the problem (5), (6) and the problem (8), (6),because we can obtain the same results of the problem (1), (2)and the problem (3), (2) by the transform (4) and the transform (7) ,respectively.This paper consists of four chapters. The first chapter is the introduction. In the second chapter, we will study the existence and uniqueness of the local generalized solution ,the local classical solution , the global generalized solution and the global classical solution for the initial value problem (5), (6) of nonlinear wave equation of fourth-order. In the third chapter, we will give the sufficient conditions of blow-up of the solution for the problem (5), (6). In the fourth chapter, we will study the existence of the global generalized solution ,the global classical solution and blow-up of the solutions for the problem (8), (6). The main results are the following:Theorem 1 Suppose that(1) s > 3/2,φ(s)∈C^[s]+1(R) andφ(0) = 0;(2) u₀(x)∈H^s(R) and u₁(x)∈H^s(R).Then the problem (5), (6) admits a unique local generalized solution u(x, t)∈C([0, T₀); H³(R)) ,where [0, T₀) is the maximal time interval.Theorem 2 Suppose that(1) s > 5/2,φ(s)∈C^[s]+1(R),φ(0) = 0,ψ(u_x)≥0 andψ(u_0x)∈L¹;(2) u₀(x)∈H^s(R) and u₁(x)∈H^s(R).(3) |φ(u_x)|≤Aψ(u_x)^1/ρ|u_x| + B,where A,B > 0,1≤ρ≤∞, whereψ(u_x) =∫₀^u_xφ(s)ds, then the problem (5), (6) admits a unique global generalized solution u(x, t)∈C([0,∞); H^s). Remark 1 Under the conditions of Theorem 2 ,if s > 5/2,the problem (5), (6) admits a unique global classical solution u(x,t)∈C²([0,∞);C²(R)) .Theorem 3 Assume thatφ(s)∈C(R), u₀(x)∈H¹(R) , u₁(x)∈H¹ (R) andψ(s)∈L¹ and there isγ> 0,which satisfies sφ(s)≤(3 + 4γ)ψ(s), (?)s∈R, then the generalized solution u(x, t) or the classical solution u(x, t) of the problem (5), (6) blows up in finite time if one of the following conditions is satisfied. (1) E(0)<0; (2) E(0) = 0 and (3) E(0)>0 and whereTheorem 4 Suppose that(1) a > 5/2, g(s)∈C^[s]+1(R), g(0) = 0,C(u_x)≥0,ζ(u_0x)∈L¹;(2) u₀(x)∈h^s(R) , u₁(x)∈H^s(R).(3) |g(u_x)|≤Aζ(u_x)^1/ρ|u_x| + B,whereA,B> 0,1≤p≤∞, whereζ(u_x) =∫₀^u_x g(s)ds, then the problem (8), (6) admits a unique global generalized solution u(x, t)∈C([0,∞); H^s(R)).Remark 2 Under the conditions of Theorem 4 ,if s > 5/2,the problem (8), (6) admits a unique global classical solution u(x, t)∈C²([0,∞); C²(R))..Theorem 5 Assume that g(s)∈C(R),u₀(x)∈H¹(R) , u_x(x)∈H¹(R),ζ(s)∈L¹ and there existsγ> 0,which satisfies sg(s)≤(3+4γ)ζ()s, (?)s∈R, then the generalized solution u(x, t) or the classical solution u(x, t) of the problem (8), (6) blows up in finite time if one of the following conditions is satisfied. (1) E(0) < 0; (2) E(0) = 0 and (3) E(0) > 0 and where,...