On The Long-time Dynamic Behaviour Of Semilinear Wave Equations

In this doctoral dissertation,we mainly study the long-time dynamic behaviour of semilinear wave equations with nonlocal weak damping and anti-damping where k and p are positive constants,l?0,?(?)Rn is a bounded smooth domain,K?L2(?×?),h?L2(?),f?C1(R)satisfies the dissipativity condition and the polynomial growth condition with subcritical exponent or critical exponent.The thesis mainly consists of three parts.In the first part(Chapter 1 and Chapter 2),we give a brief sketch of the background,the main conclusions and the preliminary knowledge of this thesis.In the second part(Chapter 3 and Chapter 4),we establish the global well-posedness,the dissipativity and the existence of global attractor when l=0 and f is of subcritical and critical growth respectively.To prove the global well-posedness,we first establish the existence and uniqueness of the strong solution and generalized solution as well as their regularity by the theorem concerning the well-posedness for the evolution equation containing an m-accretive opera-tor with local Lipschitz perturbation,along with the energy method.Then we prove the generalized solution is also a weak solution by verifying the variational equality.Note that we do not impose any restriction on the exponent p in the nonlocal damping coefficient except p?0,which creates special obstacles to prove the dissipativity and the existence of global attractor.Besides,due to the influence of the anti-damping(?)K(x,y)ut(t,y)dy,the energy is not decreasing along trajectories and thus the typical method to prove the dissipativity based on the construction of a suitable Gronwall's inequality fails in this case.To over-come these difficulties,we first construct a refined Gronwall's inequality and then use the barrier method to prove the dissipativity for this system.Afterwards,we prove that the system possesses the global attractor by the method of Condition(C)when f is of subcritical growth.When f is of critical growth,since the corre-sponding Sobolev embedding is no longer compact,we can not apply the method based on compactness to prove the existence of the global attractor.To handle this difficulty,we first establish a monotone inequality in a general inner product space,which will play a vital role in establishing the estimate of the polynomial decay of noncompactness measure of bounded subsets for the wave equation in Chapter 5;then we prove the existence of the global attractor for this system by the method of contractive function in the case of critical growth.The third part(Chapter 5)focuses on the decay rate of noncompactness mea-sure of bounded subsets for the dynamical system.As far as we know,the results on the polynomial decay with respect to noncompactness measure is proposed for the first time in the field of the infinite dimensional dynamical system.As a preliminary,in Section 5.1,we prove three lemmas on the estimate of the decay rate of nonnegative function,which are the generlization of some results by M.Nakao.Next,in Section 5.2,we establish the abstract theory on the?-decay of noncompactness measure of bounded subsets,which is an extension of the concept of exponential decay with respect to noncompactness measure put forward by J.Zhang,et al.We prove that for each dissipative dynamical system satisfying the ?-decay with respect to noncompactness measure,there exists a positively invariant compact set which attracts any bounded set B at the rate?(t-t*(B)-1),and then propose several criteria of the ?-decay with respect to noncompactness measure.In particular,theorem 5.19 approaches the critical problem via the contractive function,and theorem 5.21 gives the estimate of the polynomial decay with respect to noncompactness measure for dynamical systems under certain conditions.In Section 5.3,as an application of abstract theorem 5.19,we prove that the wave equation(0.0.2)is exponential decay with respect to noncompactness measure via Gronwall's inequality provided that l>0 and f is of critical growth.In Section 5.4,as an application of abstract theorem 5.21,we give the estimate of polynomial decay with respect to noncompactness measure for the wave equation(0.0.2)provided that l=0 and f is of subcritical gowth.