Cauchy Problems For Boussinesq Equation With Singular Integral Terms

This paper consists of five chapters.The first chapter is the introdution.In the second chapter,we will study the existence and uniqueness of the local solution to the Cauchy problem for a class of Boussinesq equations with singular integral terms.In the third chapter,we will prove the existence and uniqueness of the global solution to the problem mentioned in Chapter two by integral estimates.In the fourth chapter,we will discuss the blow-up of the solution to the problem mentioned in Chapter two.In the fifth chapter,we will get some integral estimates with Hilbert transform in the condition of small initial data,we also get the decay property of solutions,then we will prove the existence of the global solution,these are new results.In the second chapter.we study the existence and uniqueness of the local solution in the following Cauchy problem for a class of Boussinesq equations with singular integral terms.where u(x, t) denotes the unknown function, α > 0, β ≥ 0, γ>0 are constants, H is Hilbert transform,its definition isf(s) is the given nonlinear function, φ(x) and ψ(x) are given initial value functions, and subscript t,x indicates the partial derivative with respect to t, x. For this purpose,we first consider the following linear problemAfter the existence and uniqueness of the local solution to the problem (0.3),(0.4) are proved, using the contraction mapping principle we can prove the existence and uniqueness of the local solution to the nonlinear problem.The main results are the following:...