Long Time Behavior Of Solution Of Semilinear Wave Equation With A Logarithmic Source Term

Wave equation in physics,engineering,fluid mechanics,materials science and other fields has a broad application,many scholars have invested a lot of energy to study these equations of related issues,including the existence,uniqueness and stability of the solution of wave equation and instability properties become one of the important research topic.In this paper we study the asymptotic stability and exponential growth of solutions for a class of semilinear wave equations with variable coefficients with pair of source terms and memory terms.The characteristic of this paper is that the variable coefficient is introduced in front of the source term,which can describe the wave conduction under non-uniform external force.We prove the exponential decay of energy by means of energy method with appropriate multipliers.By selecting the appropriate auxiliary functional and deriving the corresponding differential inequality,we find the exponential growth of the energy of the equation under certain initial conditions.The influence of pair of source terms and memory terms on the long-term behavior of wave equation solutions is further revealed as follows:1.Long time stability of solutions.We use the multiplier method,and constructe energy functional E(t)and its differential equation E'(t)on H01(?).According to our assumptions and lemma,We can first obtain that the differential equation of the energy functional E(t)is Less than or equal to zero,so energy functional E(t)is increasing.Through Talenti-Sobolev inequality,logarithmic Sobolev inequality,Young's inequality and embedding theorem,the?u?22??u?22 and ??u?22 terms of equation are estimated.The nonnegative property of the energy functional E(t)is obtained,and proved the existence of positive number ? and ?satisfy E(t)??E(0)e-??(t)on this basis,which further shows that the energy functional E(t)is exponentially decaying,that is,the equation is asymptotically stable.2.Exponential growth of the solution.In this section,we use k0=1,k1=0 as an example,to show that the solution of the equation increases exponentially at infinity,which is not substantially different from the proof in other cases.First,we construct an appropriate functional?(t)to prove that the solution of the equation of ?u?2 has a non-zero bound under the condition of large initial values.Then,by constructing the auxiliary functional L(t)equivalent to the energy functional,and get L(t)which satisfied differential inequality is obtained,and the exponential growth of solution is proved.Now,it's worth mentioning that in the proof we're just asking for |b(x)|0 and L(t)?*in the sense that ??|u(x,t)|2 dx+??|ut(x,t)|2dx?+?,t??.