An example of nonlinear wave equation whose solutions decay faster than exponentially
type: Item
instance of: scholarly article
title: An example of nonlinear wave equation whose solutions decay faster than exponentially
zbMATH Open document ID: 0653.35062
DOI: 10.1016/0022-247X(87)90357-X
author: Mitsuhiro Nakao
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: The author studies the mixed problem: \[ u_{tt}-u_{xx}+(\partial /\partial t)f(x,u)+g(x,u) = 0 text{\quad in\quad} I\times R+, \tag{1} \] where \(I\) is a bounded interval in \(R\), with the conditions \[ u(x,0) = u_ 0,\quad u_ t(x,0)=u_ 1\quad and\quad u |_{\partial_ I}=0. \tag{2} \] Under certain assumptions for the growth of (\(\partial /\partial t)f(x,u)\) and g(x,u) the author proves the existence of a generalized solution \(u\in L^{\infty}(R\) \(+;\overset\circ H_ 1)\cap W^{1,\infty}(R\) \(+,L\) 2) of the problem (1), (2) satisfying the inequality E(u(t))\(\leq C_ 0\exp \{-\exp \lambda t\}\) for any \(t>0\). Here \(C_ 0\) is a positive constant depending on initial functions \(u_ 0\), \(u_ 1\) and E(u(t)) denotes the energy operator generated by the equation (1) and \(\lambda\) is a parameter depending of the growth conditions.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 4065540
zbMATH Keywords: mixed problem
zbMATH Keywords: growth
zbMATH Keywords: existence
zbMATH Keywords: generalized solution
zbMATH Keywords: inequality
zbMATH Keywords: energy operator
zbMATH Keywords: parameter
reviewed by: Vladimír Ďurikovič
MaRDI profile type: Publication
cites work: Growth estimates for solutions of evolutionary equations in Hilbert space with applications in elastodynamics
cites work: Some uniqueness and growth theorems in the Cauchy problem for \(Pu_{tt}+Mu_t+Nu=0\) in Hilbert space
cites work: The asymptotic behavior of solutions of second order systems of partial differential equations
cites work: Periodic solution and decay for some nonlinear wave equations with sublinear dissipative terms
cites work: Lower Bounds for Solutions of Hyperbolic Inequalities
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description: scientific article; zbMATH DE number 4065540