The Initial-Boundary Value Problem Of A Kind Of Fourth-Order Wave Equations With Nonlinear Damping And Source Terms

In this paper, the initial-boundary value problem of a kind of nonlinear fourth-order wave equations with damping and source terms is studied on the base of Beam Equation. By using the important theorem of the Sobolev space (embedding theorem) and the standard semi-group theory, local existence of the mild solution is proved. When the effect of the nonlinear damping term is stronger than the source term, By using modified energy function, according to continuation principle, the mild solution can be retarded the global solution. This paper researches the problem—the nonlinear damping and source terms affects the blowing-up of the problem. When the effect of the nonlinear damping term is weaker than the source term, by choosing the appropriate initial value and using compensating energy, the solution blows up in finite time, and the super of time span for blows-up of the solution is obtained. We can estimate whether the physical model or the mathematical model correspondingly has warp or not when the blow-up of the solution is known.Moreover, By using the method of function energy and constructing stable set, according to the potential well theory, the energy decay property of the global solution is proved.