The Cauchy Problem For A Class Of The Wave Equations With A Singular Integral Term

This paper consists of four chapters. The first chapter is the introduction, this chapter gives the problem we want to study, the source and the deduciblc process of the equation , some notations, as well as some important conclusions; In the second chapter, we will show the existence and uniqueness of the local solution to the Cauchy problem for a class of the wave equations with a singular integral term by the contractive mapping principle: In the third chapter, first we get a sufficient and necessary condition of the existence and uniqueness of the global solution, also gain some integral estimates by the energy identity, then we will study the existence and uniqueness of the global solution to the Cauchy problem mentioned in chapter two; As to the fourth chapter, first we will prove a generalized convexity lemma, then we will discuss the blow-up of the solution to the Cauchy problem mentioned in chapter two using the lemma we have proved, and give a sufficient condition of blow-up of the solution.The details arc as follows:In the second chapter, using the contractive mapping principle, we show the existence and uniqueness of the local solution to the following Cauchy problem for a class of the wave equations with a singular integral term:where u(x,t) denotes the unknown function, f(s) is a given nonlinear function,φ(x) andψ(x) are given initial functions, J > 0 is a constant, (?); subscripts t and x indicate the partial derivative with respect to t and x; H is the Hilbert operator, itsdefinition iswhere P.V. denotes the Cauchy principal value.For this purpose, at first we discuss the existence and uniqueness of the local solution to the following Cauchy problem of the corresponding linear equationand get some important estimates; then invert problem (1.1),(1.2) into an integral equation by Fourier transform, we prove the existence and uniqueness of the solution to the integral equation using the contractive mapping principle, hence we gain the existence and uniqueness of the local solution to the Cauchy problem (1.1),(1.2).The main conclusion is:Theorem 1 Suppose s≥(?) ,φ∈H^s,ψ∈H^s-2, and f∈C^[s]+1(R), f(0) = 0, then the Cauchy problem (1.1),(1.2) has an unique local solution u(x, t)∈C(0, T₀;H^s)∩C¹(0,T₀;H^s-2), where T₀ is the maximal time of the existence of the solution; Furthermore, ifthen T₀ =∞, that is to say (?) is a global solution.In Chapter three, first we get a sufficient and necessary condition of the existence and uniqueness of the global solution, and gain some integral estimates by the energy identity, then we prove the existence and uniqueness of the global solution to the Cauchy problem (1.1),(1.2).The main results are as follows:Theorem 2 Suppose (?), and let T > 0 is the maximal time of the existence of the corresponding solution u(x, t) to the Cauchy problem (1.1),(1.2), then T <∞if and only ifTheorem 3 Assume (?),and if f(u) satisfies one of the following conditions, then problem (1.1),(1.2) has an unique global solution (?). (1)F(u)=(?);(2) f'(u) is bounded below, i.e. there exists a constant A₀, such that (?)_u∈R, f'(u)≥A₀.In Chapter four, at first, we prove a generalized convexity lemma, then we discuss the blow-up of the solution to the problem (1.1).(1.2) in a finite time by the lemma we have proved, and give a sufficient condition of blow-up of the solution.The main results are as the following:Theorem 4 Suppose that a positive, twice differentiale functionΦ(t) satisfies the following inequality on t≥0whereα> 0,β≥0, A≥0 and B≥0 are constants. IfthenΦ(t) tends to infinity as t tends to T≤T^*. whereTheorem 5 Suppose (?), and there exists a constantα> 0 such that (?)_u∈R,Then the solution to the Cauchy problem(1.1),(1.2)blows-up in a finite time if and only if the initial values satisfy one of the following conditions:(1) E(0) < 0;(2)E(0) =0,(?);(3)E(0) >0,(?).