Exponential Energy Decay Estimate Of Several Wave Equation With Specific Conditions

Today, interdisciplinary cross and mutual compatibility among each course are more and more obvious.Mathematics, as the basis of natural science, is not exceptional and grad-ually plays an important role in physics and engineering.Integro-differential equations has always been an important branch of mathematics,so the existence and uniqueness of solution determined the real-life problem whether can get a solution.Furthermore,the performance of the solution affected all aspects of the practical problems.In this paper,wave equation is mainly regard as the research object,on the background of practical problems,on the basis of many scholars’research conclusion,we make further research about attenuation situation of three specific wave equation.The first chapter as the introduction of this paper, mainly introduced the body content succinctly.In chapter2,on the basis of conclusions about exponential energy decay rates of cou-pling vibration system that Najafi.M,Lions J L and other scholars have given,we studied the exponential stability of the coupled wave equation.we mainly make the improvement on the following equation: Let Ωu and Ωv,with respect to the vector field r(x)=[r1(x),…,rn(x)]∈C2(Ω2∪Ωv), be such that the system satisfies the following boundary conditions: where v(x) is the external normal function at the point x∈Γi；i=0,1. The boundary conditions can now be prescribed as follows:The governing equations prescribing the stabilization and control of a system of two hyper-bolic equations coupled in parallel are: Let ΩuandΩv,with respect to the vector field r(x)=[r1(x),…,rn(x)]∈C2(Ωu∪Ωv), be such that the system satisfies the following boundary conditions: where v(x) is the external normal function at the point x∈Γi,i=0,1. The boundary conditions can now be prescribed as follows:The third chapter mainly studies stabilization of nonlinear vibrations of a flexible struc-ture. On the basis of Gorain,Boso,Horns’ study about exponential energy decay estimates for the solutions of n-dimensional Kirchhoff type wave equation,that is: we make promotion about above equation. In this paper, we apply a velocity feedback control only on a part of the boundary under mixed boundary conditions and get equation as follows: Finally,we establish the uniform decay of solution by a direct method,with an explicit form of exponential energy decay estimate.In the fourth chapter,we mainly studied the solutions of internally damped wave equa-tion in a bounded domain. Chen,Lagnese,Lasiecka,Lions and so on have given some conclu-sion about undamped wave equation.Such as:The authors have obtained energy decay estimate of above equation,that is:.E(t)<Me-βE(0)(?)t> O.In this chapter,we obtained somewhat faster energy decay rate than above equation when internal damping δut is present in the left hand side of above equation.That is: we also established the uniform decay of solution and proved that the system is consistently stable.