The Studies On Some Problems For Semilinear Mindlin-Timoshenko Beam Equations

In this paper, the following beam equations are discussed. The thesis aims to study the nonexistence and existence of global solutions for system (1), as well as the energy decay estimate of global solutions for system (2).In chapter two, the potential well theory is introduced and the related theorems are proved.In chapter three, the existence theorem of global solutions for system (1) is obtained with the help of potential well theory and Galerkin method.Theorem1Let u0,v0∈V, u1,v1∈V and f(u),g(v) satisfy (P2). If [u0,v0]∈W and E(0)<d, then there exist strong solutions for system (1) such that u,v∈L∞(0,T;V), ut,vt∈L∞(0,T;V),utt,vtt∈L2(Q).Combining the concavity method, chapter four proves that the solution blows up in finite time.The results are following:Theorem2Letu, v be local solutions for systems (1).If E(0)<0,then the solutions blow up in finite time.Theorem3Let u, v be local solutions for systems (1). If E(0)=0and∫0l(u0u1+v0v1)dx>0, then the solutions blow up in finite time. Theorem4Let u, v be local solutions for systems (1). If0<E(0)<d and∫0l(u0u1+v0v1)dx>0, then the solutions blow up in finite time.In the last chapter, by using M.Nakao difference inequality, a decay estimate of global solutions for system (2) is derived.Theorem5Let u,v be solutions for system (2), then there exist positive constants λ, K such that E(t)≤Ke-λt,for t≥0.