Longtime Dynamics Of Three Kinds Of Nonlinear Wave Equations With Fractional Damping

In this thesis,we study the well-posedness and longtime behavior of three kinds of nonlinear wave equations with fractional damping.(i)In chapter 2,for the semilinear wave equation with gentle dissipation with ??(0,1/2),the main results are concerned with the relationships among the growth exponent p of nonlinearity f(u)and the well-posedness and longtime behavior of solutions.We find a new critical exponent p*=N+4?/(N-2)+ of the unique-ness of solutions.In the subcritical case,i.e.,1?p<p*,we prove the global well-posedness of the solutions in phase space E1=?1×L2,and prove that the equation is like parabolic,i.e.,its weak solution is of higher global regularity(rather than higher partial one as usual).Moreover,using the stability in the weaker topology space E1/2-? and the regularity in the stronger topology space E1+?,we show the existence and the upper semicontinuity depending on dissi-pative exponent ? of the finite dimensional global attractors in natural energy space E1.by applying the quasi-stability in the weaker topology space and the recover of compactness,we prove the existence of exponential attractor.At last,In the critical and supercritical cases(p*?p<N+2/N-2),when the uniqueness of energy solutions is lost,we prove the existence of limit solutions and a weak global attractor of the subclass L of limit solutions for Eq.(0.0.1).(ii)In chapter 3,for the quasilinear wave equations with structural damping with ??(1/2,1),we study the well-posedness and the longtime dynamics of the equations(0.0.2)with supercritical nonlinearities.The main results are con-cerned with the quasilinear term ?·((?))and the nonlinearities f(u)with supercritical growth.Under the rather mild conditions,the well-posedness and the existence of the global attractor(rather than the weak ones)are established in natural energy space H=H01 ?Lp+1×L2.These results show that even for the supercritical nonlinearities,the higher global regularity and the longtime dynam-ics of the solutions are like parabolic because of the effectiveness of the structural damping.We propose a new criterion on the existence and stability of a family of exponential attractors depending on the perturbation parameters.Moreover,by applying this criterion to the equations(0.0.2),we construct a family of expo-nential attractors Aexp? and show their stability with the exponential attractor of the semilinear wave equation.(iii)In chapter 4,we study the global well-posedness and the longtime dy-namics for the Kirchhoff models with structural damping utt-??·?(?u)+?(-?)?ut+?2u+h(ut)+f(u)=g,with ??[1,2).The main results are concerned with the nonlinearities ?·?'(?u)and f(u)with nonlinear damping h(ut).We are further concerned with the existence and upper semicontinuity of global attractors in natural energy space?= H2 ?H01×L2.These results show that the higher global regularity and the longtime dynamics of the solutions are like parabolic because of the effectiveness of the structural damping.Moreover,by applying the criterion given in chapter three to the equations(0.0.3),we construct a family of exponential attractors and show their stability depending on the perturbation parameters.At last,we prove that solution semigroup has in natural energy space an optimal global attractor and an optimal exponential attractor,whose compactness,boundedness of the fractional dimension and the attractiveness are in the regularized space ??.