Longtime Dynamics Of Kirchhoff Type Wave Equations With Damping

In this paper,we mainly research the global well-posedness and longtime dynamic behavior of three kinds of autonomous Kirchhoff type wave equations with damping.1.The Kirchhoff type wave equations with strong damping on unbounded domain:2.The Kirchhoff type wave equations with variable coefficients and strong damping on unbounded domain:3.The Kirchhoff type wave equations with Dirichlet boundary condition and structural damping on bounded domain:Combining the energy method with the monotone technology,we obtain the global well-posedness of solutions for?0.0.1?in the natural energy space H=H¹?R^N?×L²?R^N?.On account of a new cut-off function,we obtain the tail cut-off estimate of solutions,which,combining with the recently developed quasistable estimate,overcomes the essential difficulties “both the Sobolev embedding on unbounded domain and the critical growth of nonlinearity f cause the lack of compactness”;Based on these methods,we establish the existence of the finite dimensional global attractor and the exponential attractor in phase space H.For the Kirchhoff type equations with variable coefficients and strong damping on unbounded domain,the global well-posedness is established in weighted phase space when the growth rate H of nonlinearity f)is up to the supercritical case,that is, .By the weak quasi-stable estimate and the compensated compactness method,the finite dimensional global attractor and the exponential attractor are established in ?in the sense of the partially strong topology.In particular,in non-supercritical case,i.e.,,the finite dimensional global attractor and the exponential attractor are established in with the strong topology,and the global attractor in the sense of the strong topology coincides with the partial strong one.We find a new critical exponent = +4?-4?+for?0.0.3?with Dirichlet boundary condition and structural damping on bounded domain,and is also the exponent of the uniqueness of solutions.When 1 ? p <p_a,we prove the wellposedness of the solutions in phase space H =(H₀¹? L^P+1)×L²,and prove that the solutions is of the characters of the parabolic equation.Moreover,making use of the quasi-stability in the weaker topology and the recover of compactness,we show the existence of the finite dimensional global attractor and the exponential attractor.In the supercritical case,that is, ,we get the existence of limit solutions for Eq.?0.0.3?and construct the subclass G of limit solutions.At last,we prove that the subclass G possesses a weak global attractor by a constructive approach.