Asymptotic stability for some dissipative models of plate equations

Author(s)

Marcio Antonio Jorge da Silva

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2012-03-13

Examining board members

Ma To Fu; Alexandre Nolasco de Carvalho; Valéria Neves Domingos Cavalcanti; Marcone Corrêa Pereira; Jaime Edilberto Munoz Rivera

• Advisor: Ma To Fu

Abstract

In this work we study some questions concerning with existence, uniqueness, continuous dependence, continuity, rates of decay and asymptotic behavior of solutions for a class of linear and nonlinear plate equations. In the first chapter we review some concepts and collect a series of results provided from general theory of functional analysis, linear semigroups and attractors which will be applied throughout this thesis. In the next two chapters we discuss a damped plate equation of fourth order with nonlinear perturbations of the lower order of p-Laplacian type and locally Lipschitz, and a memory term. In the second chapter we prove the exponential stability of energy corresponding to the homogeneous problem with memory of second order. Then in the third chapter we establish some results that allow us to prove the existence of a global attractor with finite fractal dimension for dynamical system associated to the problem with relative displacement history which is equivalent to the original problem. Finally, in the fourth chapter we deal with a viscoelastic Mindlin-Timoshenko plate model of second order. At this moment we consider essentially two cases. The first one when the system is fully damped, then when the system is partially damped. In the first case we show that the semigroup associated to the Mindlin-Timoskenko system is analytic, which in particular implies exponential decay. In the second case we show that such semigroup loses exponential decay, also loses analyticity. However, we prove in this last case that the solutions have decay of polynomial type (AU)

• FAPESP's process: 08/00123-7 - Asymptotic stability of nonlocally defined evolution equations.
• Grantee: Marcio Antonio Jorge da Silva
• Support Opportunities: Scholarships in Brazil - Doctorate