Initial Boundary Value Problem For A Damped Nonlinear Hyperbolic Equation

This paper consists of four chapters.The first chapter is the introdution.In the second chapter,we will study the existence and uniqueness of the local solution to the initial boundary value problem for a damped nonlinear hyperbolic equations in the three-dimensional.In the third chapter,we will prove the first global nonexistence theorem of the solutin to the problem mentioned in Chapter 2 and give an example. In the fourth chapter,we will prove the second global nonexistence theorem of the solutin to the problem mentioned in Chapter 2.In the second chapter,we study the following initial boundary value problem for a damped nonlinear hyperbolic equations (in the three-dimensional)with the boundary value conditionsand the initial value conditionswhere u(x, t) denotes the unknown function, Ω is a bounded domain in Rⁿ with smooth boundary (?)Ω, k₁ and k₂ are two positive constants, V denotes the gradient operator, (?)² = Δ denotes Laplacian operator, (?)⁴ = Δ² denotes the biharmonic operator, g(s) is the given nonlinear function and subscript t indicates the partial derivative with respect to t. For this purpose,we first give two lemmas, then prove the existence and uniqueness of local generalized solution for the problem (0.1)-(0.3) by the Galerkin method and compactness theorem. Next we give the third lemma and prove the existence and uniqueness of local classical solution for the problem (0.1)-(0.3) by the same method. The main results are the fowllowing:...