Stability Of Regular Attractors For The Kirchhoff Type Equations With Damping

In this dissertation,we mainly investigate the well-posedness,the existence and the stability of regular attractors for three kinds of Kirchhoff type equations with damping.1.In chapter 3,for the Kirchhoff wave equation with strong damping???with???? [0,1],we study the well-posedness in phase space ? =H₀¹×L₂ when the growth exponents of nonlinearities h?s?and g?s? are of critical(1 ? q,p ??N +2?/??N-2?⁺?.We prove that its weak solution is of higher partial regularity,i.e.,the equation is like parabolic.By the Lipschitz continuity in ? and the regularity in H₀¹× L₀¹,we show the existence of global attractors which have finite fractal dimension and their upper semicontinuity on the perturbation parameter???in ?.Using the quasi-stability in ?,we get the existence of exponential attractors and their stability on ???.2.In chapter 4,for the Kirchhoff wave equation with structural damping???with???? [0,1],???1/2,1?,we obtain the existence of unique weak solution in phase space ? = H₀¹× L² when the growth exponents of the nonlinearities h?s?and g?s? satisfy the following conditions: 1 ? q ? 3,N = 1;1 ? q <q_???N +??/?N-??,N ? 2,1 ? p ??N +2?/??N-2?⁺?.On account of the weak solution is of higher global regularity?rather than higher partial one?,we establish the weak solution is exactly the strong solution.Combining the Lipschitz continuity in the weaker topology space with the regularity in the higher topology space,we prove the existence of global attractors and their upper semicontinuity on the perturbation parameter ? and ???,respectively.Based on the quasi-stability in the weaker topology space,the existence of exponential attractors and their stability on ??? can be obtained.Moreover,we show the existence of regular attractors whose compactness,attractiveness and boundedness of the fractional dimension are in the regularized space.3.In chapter 5,for the extensible beam equation with gentle damping???with ???0,1?,we are concerned with the well-posedness when the growth exponent of the nonlinearity f?s? satisfies: 1 ? p <?N +4??/??N-4?⁺?(particularly,when f is the real source term,???= S^m/2,s ? R⁺,1 ? p <m+ 1,N ? 2;1 ? P ??N +2?/?N-2?and P <m+ 1,N ? 3).We prove the existence of unique weak solution in phase space ? = V₂× L².By applying the higher global regularity,we can obtain the weak solution is exactly the strong one.Using the Lipschitz continuity in the weaker topology space and the regularity in the higher topology space,we prove the existence of global attractors and their upper semicontinuity on the perturbation parameter ?.Taking advantage of the quasi-stability in the weaker topology space,we can get the existence and the stability on ? of exponential attractors.At last,we prove the existence of regular attractors.