The Cauchy Problem For A Class Of Damped Nonlinear Wave Equations Of Fourth-order

In this paper, we are concerned with the following Cauchy problem for a class of damped nonlinear wave equation of fourth-order:where a₁,a₂,a₃＞0 are constants, v(x, t) denotes the unknown functionφ(s),ψ(s) and h(s) are given nonlinear functions,,and subscripts x and t indicate the partial derivative with respect to x and t, respectively. Equation (1) includes many mathematical models,which describe physical phenomena. For simplicily of discussion,we make a change of variablesthen the Cauchy problem (1), (2) becomesIn 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 (4), (5) ,because we can obtain the same results of the problem (1), (2) by the transform (3) .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 and the local classical solution for the Cauchy problem (4), (5) of damped nonlinear wave equation of fourth-order .In the third chapter,we will study the existence and uniqueness of the global generalized solution and the global classical solution for the Cauchy problem (4), (5) of damped nonlinear wave equation of fourth-order . In the fourth chapter, we will study the blow-up of the solution for the problem (4), (5). The main results are the following:Theorem 1 Suppose that (1)s＞3/2,f,g,h∈C^[s]+1(R),f(0)=0,g(0)=0,h(0)=0; (2)u₀∈H^s(R),u₁∈H^s(R).then the problem (4), (5) admits a unique local generalized solution u∈C([0,T₀);H³(R)), where [0. T₀) is the maximal time interval.Theorem 2 Suppose that(1)s＞5/2,f,g,h∈C^[s]+1(R),f(0)=0,g(0)=0,h(0)=0,|f'(s)|≤C₀,C₁≥(u_x)≥0, H(u)≥0andG₁(u_0x),H(u₀)∈L¹(R), where C₀＞0 are constant:(2)u₀∈H^s(R),u₁∈H^s(R); (3)|g(u_x)|≤A₁G₁(u_x)^?|u_x|+B₁and|h(u)|≤A₂H(u)^?|u|+B₂,whereA_i,B_i＞0,, 1≤ρ_i≤∞(i=1,2)are constants,then the problem (4), (5) admits a unique global generalized solution u∈C([0,∞);H^s(R)).Remark Suppose that u∈C([0,∞);H^s) is a generalized solution of the problem (4), (5),if s＞5/2,the problem (4), (5) admits a global classical solution u∈C²([0,∞);C_B²(R)).Theorem 3 Assume that f(s) = 0 , g,h∈C(R),u₀,u₁∈H¹(R),G1,H∈L¹(R) and there is a＞0,which satisfiesthen the generalized solution u or the classical solution u of the problem (4), (5) blows up in finite time if one of the following conditions is satisfied: where...