On The Long-time Dynamical Behavior Of A Class Of Extensible Beams Equation With The Nonlocal Weak Damping

In this doctoral dissertation,we study the well-posedness,the long-time behavior of solutions and the geometry to the following extensible beam equation with nonlocal weak damping(?)(4)where ?(?)Rn is a bounded domain with smooth boundary(?)?,m(s)is a nonlocal coefficient,h ? L2(?)is an external force term,f(u)is a source term,and the nonlinear function k is a polynomial function,and it satisfies the following forms k(s)=?0+(?)?ispi,k(0)=0,(?)s ?[0,?),?0? 0,?k>0,0?pi<pk<?,(5)where the coefficients {?i,1 ?i?k-1} may be negative.In Chapter 3,we study the well-posedness and the existence of global attrac-tor for a class of extensible beams equation(4)with the nonlocal weak damping k(?ut?)ut=?ut?put,p>0(i.e.k(s)=sp,?i=0,0?i<k-1)when the growth exponent of the nonlinearity f(u)is up to the subcritical case.In this chapter,inspired by the monotone inequality of the p-Laplacian operator in the European space given by Simon[142]and Perai[121],we first generalize this monotone inequality to Hilbert space,So that the strong monotonicity of the damping term is obtained,which is useful to prove the well-posedness,dissipativity and the asymptotic smooothness to problem(4).Unlike many other literatures,it is difficult for us to use the Fatou-Galerkin method to prove the well-posedness of the problem(4)is mainly due to that for the solution of the approximation equa-tion,through energy estimation,we can only get unt in boundedness in the L2(?)norm,thus obtaining the weak convergence of unt in L2(?),which cannot make the nonlocal coefficient ?unt?p converges to the same limit.Finally,similar to the energy reconstruction method used by Chueshov and Lasiecka[27],we obtain the asymptotic smoothness of the semigroup.In[27],the energy reconstruction method proved the existence of the global attractor of the equation(4)when the growth index of the nonlinear damping g(ut)is subcritical and subcritical nonlinear term F.The main reason why we need to use energy reconstruction method is because of the velocity ut is very small,the nonlocal damping ?ut?put is weaker than the linear damping ut,so it is difficult to obtain Gronwall's in-equality when we use the usual energy estimates.It is worth mentioning that the p in our nonlocal damping term does not have any upper limit,which brings us many difficulties to prove the dissipation and the asymptotic smoothness of the problem(4).In Chapter 4,we devoted to establishing the existence of global attractor of extensible beams equation with the nonlocal weak damping(4)in the critical case.Since the damping term k(?ut?)ut in(4)we consider in this chapter is a more general case,and the coefficients {?i,1?i?k-1} of the nonlocal damping k(?ut?)ut may be negative,which will have the effect of anti-damping and weaken the impact of damping.Therefore,these will bring some difficulties in dealing with the dissipation and compactness of the semigroup.Compared with the subcritical case,the main difficulty brought by the critical case is that it no longer has the compactness embedding property.Therefore,we consider using the contraction function method to prove the asymptotical smoothness of the semigroup,and then obtain the existence of global attractor for the problem(4).Although the contraction function is an effective method to deal with such problems as critical nonlinear terms,it is very troublesome to construct and verify the compression function.In particular,because the non-local damping term is k(?ut?)ut and pi is not the upper bound,therefore,there are many difficulties in the analysis and calculation of the dissipation of semigroups and progressive smoothness.In Chapter 5,we are interested in the study of the existence of a finite-dimensional global attractor and the exponential attractor for extensible beams equation(4)with the nonlocal weak damping k(?ut?)ut.And the damping k(?ut?)ut satisfies the following forms k(0)>0,k'(s)>0,k(s)=?0+(?)?ispi(6)where(?)s ?[0,?),?k>0,0?pi<pk<?,and the coefficients {?i,1?i ?k-1} may be negative which has the effect of anti-friction,that is,it will have the effect of weakening the damping.Therefore,these will have substantial difficulties in dealing with the dissipation of the semigroup.It is worth noting that k(0)>0 in(6)means the nonlocal damping coefficient is non-degenerate,which is very helpful for us to study the properties of attractors.In this chapter,we use the quasi-stability method to deal with the long-term behavior of the problem.We obtain the stable inequalities through effective energy estimation and then prove the existence of the finite-dimensional global attractor mathcal A when the growth exponent of the nonlinearity f(u)is up to the subcritical,that is,(?)>0 if 1?n ?4 or 0<(?)<4/n-4 if n? 5,.However,t is difficult to obtain the finiteness of the dimension of the global attractor by using the energy estimation method in the case of the critical nonlinearity.The main obstacle here is the lack of compactness.Then,to overcome the difficulties,we improve the regularity of f and make use of the different energy estimation methods to prove the finite-dimensional global attractor when the growth exponent of the nonlinearity f(u)is up to the critical,that is,(?)>0 if 1 ?n ?4 or 0<(?)?4/n-4 if n? 5,Finaly,the exponential attractor Aexp of the dynamical system(H,St(t))is obtained.