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On solvability of the dissipative Kirchhoff equation with nonlinear boundary damping

Zai-Yun Zhang|Jian-Hua Huang
대한수학회 (The Korean Mathematical Society)|2014.01|대한수학회보 (Bulletin of the KMS)|제51권 제1호|pp. 189-206

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참고문헌

키워드

global existenceboundary dampingdissipative Kirchhoff equationGalerkin approximation
주제분류수학

초록

In this paper, we prove the global existence and uniqueness of the dissipative Kirchhoff equation utt −M(∥∇u∥2)△u + αut + f(u) = 0 in Ω [0,∞), u(x, t) = 0 on Γ1 × [0,∞), αu/αu + g(ut) = 0 on Γ0 × [0,∞), u(x, 0) = u0, ut(x, 0) = u1 in Ω with nonlinear boundary damping by Galerkin approximation benefited from the ideas of Zhang et al. [33]. Furthermore,we overcome some dif- ficulties due to the presence of nonlinear terms M(∥∇u∥2) and g(ut) by introducing a new variables and we can transform the boundary value problem into an equivalent one with zero initial data by argument of compacity and monotonicity.

Abstract

In this paper, we prove the global existence and uniqueness of the dissipative Kirchhoff equation utt −M(∥∇u∥2)△u + αut + f(u) = 0 in Ω [0,∞), u(x, t) = 0 on Γ1 × [0,∞), αu/αu + g(ut) = 0 on Γ0 × [0,∞), u(x, 0) = u0, ut(x, 0) = u1 in Ω with nonlinear boundary damping by Galerkin approximation benefited from the ideas of Zhang et al. [33]. Furthermore,we overcome some dif- ficulties due to the presence of nonlinear terms M(∥∇u∥2) and g(ut) by introducing a new variables and we can transform the boundary value problem into an equivalent one with zero initial data by argument of compacity and monotonicity.

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관련 논문AI추천

STABILIZATION FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH A NONLINEAR SOURCE김대욱영남수학회 · East Asian Mathematical Journal2016 · 제32권 제1호Exponential Decay for the Solution of the Viscoelastic Kirchhoff Type Equation with Memory Condition at the Boundary김대욱영남수학회 · East Asian Mathematical Journal2018 · 제34권 제1호GLOBAL EXISTENCE AND UNIFORM DECAY OF COUPLED WAVE EQUATION OF KIRCHHOFF TYPE IN A NONCYLINDRICAL DOMAIN하태갑대한수학회 · 대한수학회지2017 · 제54권 제4호ASYMPTIOTIC BEHAVIOR FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH AN INTERNAL TIME-VARYING DELAY TERM김대욱영남수학회 · East Asian Mathematical Journal2016 · 제32권 제3호EXPONENTIAL STABILITY FOR THE GENERALIZED KIRCHHOFF TYPE EQUATION IN THE PRESENCE OF PAST AND FINITE HISTORY김대욱영남수학회 · East Asian Mathematical Journal2016 · 제32권 제5호General decay for a semilinear wave equation with boundary frictional and memory conditions박선혜대한수학회 · 대한수학회보2014 · 제51권 제3호

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