| 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 solution and the global solution for the Cauchy problem of a class nonlinear wave equation of sixth order. In the third chapter, we will study blow up of solution to the Cauchy problem for the above mentioned equation . In the fourth chapter, we will study the existence of the local generalized solution for the initial boundary value problems for a class nonlinear wave equation of sixth order and give sufficient conditions of blow up of solution.In the second chapter ,we study the following Cauchy problem for a Class of nonlinear wave equation of sixth order:where v(x, t) denotes the unkown function,a > 0 is a constant,g(s) is a given nonlinear function, v0(x) and v1(x) are given initial value functions, subcripts x and t indicate partial derivatives.For simplicity ,by the scaling tranformationthe equation (1) can be written as Without loss of generality,we will study the following Cauchy problemwhere u(x,t) denotes the unknown function,φ(s) is a given nonlinear function, u0(x) and u1(x) are given initial value functions. we will prove the existence and the uniqueness of the local solution for the Cauchy problem (3),(4)and give sufficient conditions of the existence of the global solution. The main results are the following:Theorem 1 Suppose that s > 1/2, u0∈Hs, C[s]+1(R) andφ(0) = 0,then the problem (3),(4)has a unique local solution u∈C2([0,T0); Hs),where [0,T0)is the maximal time interval of existence of u. Moreover,ifthen T0 =∞.Theorem 2 Suppose that s > 1/2,u0∈Hs,u1∈Hs,φ∈C[s］+1(R),φ(0) = 0 and [0, T0) is the maximal time interval of the corresponding solution u(x, t) to the Cauchy problem(3),(4).IfwhereM2 > 0 is a constant,then T0 =∞.Theorem 3 Assume that s≥1, u0∈Hs, u1∈Hs,φ∈C[s]+1(R),φ(0) = 0, L2 andΦ(u0)∈L1, ifΦ(s)≥0 orφ'(s) is bounded blow, i.e. ,there is a constant C0 such thatφ'(s)≥C0, (?)s∈R,then the Cauchy problem (3),(4) has a unique global solution u∈C2([0,∞);Hs). where and , FandF-1denote Fourier transformation and inverse transformation in R,respectively.Remark 1 under the conditions of Theorem 3, if s > 9/2 ,the Cauchy problem (3),(4) admits a unique global classical solution.In the Chapter 3,the blow up of solution to the Cauchy problem (3),(4) are proved by means of the concavity method. The main results are the following: Theorem 4 Assume thatφ(s)∈C(R), and ,and there exists constantδ> 0 such thatThen the solution u(x, t) of the Cauchy problem (3),(4) blows up in finite time if one of the following conditions holds:(1)E(0) < 0;(2)E(0) = 0,(3)E(0) > 0,In the fourth chapter,we discuss the following nonlinear wave equationwith the initial bounardy value conditionsor with the initial bounardy value conditionswhere a > 0 is constant,φ(s) is a given nonlinear function,φ(x) andψ(x) are given initial value functions. We will prove the existence of the local generalized solutions for the initial bounardy value problem(5)-(7)or(5),(8),(9) and the sufficient conditions of blow up of solution are given.The main results are the following:Theorem 5 Suppose that , then the problem (5)-(7) admits a generalized local solution u(t)∈ W2,∞([0,T];S), where 0 < T < T0,[0,T0) is the maximal time interval of exitence of u(x, t).Moreover, ifThen T0 =∞.Theorem 6 Assume thatφ(s) is a concave function ,φ(0) = 0, y0 = ,andthen the solution of problem (5)-(7) must blow up in finite time t≤T,i.e.,Theorem 7 Assume thatφ(s) is a convex even function ,φ(0) = 0, y0 = ,andthen the soluton of problem (5)-(7)must blow up in finite time t≤T1, i.e.,As for problem (5),(8),(9) we have the following results :Theorem 8 Assume that ,then the problem(5),(8),(9)admits local generlized solution u(t)∈W2,∞([0, T],D), where 0 < T < T0, [0, T0) is the maximal time interval of exitence of u(t). Moreover,ifThen T0 =∞.Theorem 9 Suppose that the following conditons hold : Then the solution of the problem (5),(8),(9)must blow up in finite time ti.i.e.,where...
|
PDF Full Text Request