Premium
Decay rates for semilinear wave equations with vanishing damping and Neumann boundary conditions
Author(s) -
Luo JunRen,
Xiao TiJun
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6733
Subject(s) - mathematics , neumann boundary condition , exponent , mathematical analysis , wave equation , polynomial , boundary (topology) , dirichlet boundary condition , nonlinear system , boundary value problem , dirichlet distribution , quotient , pure mathematics , physics , philosophy , linguistics , quantum mechanics
The paper is concerned with the semilinear wave equations with time‐dependent damping γ ( t )= α /(1+ t ) ( α >0), under the effect of nonlinear source f behaving like a polynomial, and subject to Neumann boundary conditions. Constructing appropriate auxiliary functions, we obtain an explicit uniform decay rate estimate for the solutions of the equation in terms of the exponent of f , when α is large enough. On the other hand, via a new hyperbolic version of Dirichlet quotients, we show that the upper estimate is optimal in some case, which implies the existence of slow solutions.