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    188宝金博页面版: Existence, blow-up and exponential decay for a nonlinear Love equation associated with Dirichlet conditions

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    内容提示： 61 (2016) APPLICATIONS OF MATHEMATICS No.2, 165–196EXISTENCE, BLOW-UP AND EXPONENTIAL DECAYFOR A NONLINEAR LOVE EQUATIONASSOCIATED WITH DIRICHLET CONDITIONSLe Thi Phuong Ngoc, Nha Trang City,Nguyen Thanh Long, Ho Chi Minh City(Received September 2, 2015)Abstract. In this paper we consider a nonlinear Love equation associated with Dirichletconditions. First, under suitable conditions, the existence of a unique local weak solution isproved. Next, a blow up result for solutions with negative initial energy i...

    文档格式：PDF | 页数：32 | 浏览次数：546 | 上传日期：2018-03-15 21:15:33 | 文档星级：
    61 (2016) APPLICATIONS OF MATHEMATICS No.2, 165–196EXISTENCE, BLOW-UP AND EXPONENTIAL DECAYFOR A NONLINEAR LOVE EQUATIONASSOCIATED WITH DIRICHLET CONDITIONSLe Thi Phuong Ngoc, Nha Trang City,Nguyen Thanh Long, Ho Chi Minh City(Received September 2, 2015)Abstract. In this paper we consider a nonlinear Love equation associated with Dirichletconditions. First, under suitable conditions, the existence of a unique local weak solution isproved. Next, a blow up result for solutions with negative initial energy is also established.Finally, a suf f i cient condition guaranteeing the global existence and exponential decay ofweak solutions is given. The proofs are based on the linearization method, the Galerkinmethod associated with a priori estimates, weak convergence, compactness techniques andthe construction of a suitable Lyapunov functional. To our knowledge, there has been nodecay or blow up result for equations of Love waves or Love type waves before.Keywords: nonlinear Love equation; Faedo-Galerkin method; local existence; blow up;exponential decayMSC 2010: 35L20, 35L70, 35Q74, 37B251. IntroductionIn this paper, we consider the following nonlinear Love equation with initial con-ditions and homogeneous Dirichlet boundary conditionsu tt − u xx − u xxtt − λ 1 u xxt + λu t = F(x,t,u,u x ,u t ,u xt ) (1.1)−∂∂x [G(x,t,u,u x ,u t ,u xt )] + f(x,t),0 < x < 1, 0 < t < T,u(0,t) = u(1,t) = 0, (1.2)u(x,0) = e u 0 (x), u t (x,0) = e u 1 (x), (1.3)The research has been supported by Vietnam’s National Foundation for Science andTechnology Development (NAFOSTED) under Project 101.02-2015.06.165

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