The existence and decay of solutions of a damped Kirchhoff–Carrier equation in Banach spaces

• Published in: Nonlinear analysis 2010-10, Vol.73 (7), p.2101-2116
• Main Authors: Carvalho, R.R., Milla Miranda, M.
• Format: Article
• Language: English
• Subjects: Approximation; Banach space; Decay; Decay of solution; Derivatives; Energy methods; Exact sciences and technology; Functional analysis; Initial value problems; Kirchhoff–Carrier equation; Local and global solutions; Mathematical analysis; Mathematics; Real numbers; Sciences and techniques of general use
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2010.05.038

This paper is concerned with the study of the existence and decay of solutions of the following initial value problem: (∗) | B u ′ ′ ( t ) + M ( ‖ u ( t ) ‖ W β ) A u ( t ) + ( 1 + k ( t ) ‖ u ( t ) ‖ D ( S α + 2 ) β ) A u ′ ( t ) = 0 , t > 0 u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 , where V is a Hilbert space with dual V ′ ; A and B symmetric linear operators from V into V ′ with 〈 B v , v 〉 > 0 , v ≠ 0 , and 〈 A v , v 〉 ≥ γ ‖ v ‖ V 2 , γ > 0 ; S a restriction of the operator A ; W a Banach space; M ( ξ ) the real function M ( ξ ) = m 0 + m 1 ξ with m 0 > 0 and m 1 ≥ 0 real numbers; k a positive function and α , β real numbers with α ≥ 0 and β > 1 . The successive approximation method, the characterization of the derivative of M ( ‖ u ( t ) ‖ W β ) and the Arzela–Áscoli Theorem allow us to obtain a local solution of (∗). The global solution follows by the prolongation method of solutions. The exponential decay of the solution is derived by the perturbed energy method.