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Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions
Author(s) -
Liu Wenjun,
Chen Kewang
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400343
Subject(s) - mathematics , boundary (topology) , mathematical analysis , energy (signal processing) , nonlinear system , boundary value problem , differential (mechanical device) , galerkin method , differential inclusion , physics , quantum mechanics , statistics , thermodynamics
In this paper, we prove the existence and general energy decay rate of global solution to the mixed problem for nondissipative multi‐valued hyperbolic differential inclusionsu ′ ′ − div a ∇ u + h ( ∇ u ) − Δ u ′ + Ξ + f ( u ) = 0 with memory boundary conditions on a portion of the boundary and acoustic boundary conditions on the rest of it. For the existence of solutions, we prove the global existence of weak solution by using Galerkin's method and compactness arguments. For the energy decay rates, we first consider the general nonlinear case of h satisfying a smallness condition, and prove the general energy decay rate by using perturbed modified energy method. Then, we consider the linear case of h : h ( ∇ u ) = − ∇ ϕ · ( a ∇ u ) and prove the general decay estimates of equivalent energy.