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Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions of 2 N ‐point type
Author(s) -
Anh Triet Nguyen,
Thi Phuong Ngoc Le,
Pham Ngoc Dinh Alain,
Thanh Long Nguyen
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.6773
Subject(s) - mathematics , mathematical analysis , contraction principle , exponential growth , nonlinear system , type (biology) , contraction (grammar) , boundary value problem , function (biology) , wave equation , banach space , contraction mapping , fixed point theorem , mathematical physics , physics , quantum mechanics , medicine , ecology , evolutionary biology , biology
This paper is devoted to the study of a nonlinear wave equation with initial conditions and nonlocal boundary conditions of 2 N ‐point type, which connect the values of an unknown function u ( x , t ) at x = 1,x = 0, x = η i ( t ) , and x = θ i ( t ), where 0 < η 1 ( t ) < η 2 ( t ) < … < ηN − 1( t ) < 1 , 0 < θ 1 ( t ) < θ 2 ( t ) < … < θN − 1( t ) < 1 , for all t ≥ 0. First, we prove local existence of a unique weak solution by using density arguments and applying the Banach's contraction principle. Next, under the suitable conditions, we show that the problem considered has a unique global solution u ( t ) with energy decaying exponentially as t → + ∞ . Finally, we present numerical results.