Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 263, Issue 5, Pages 2375-2418Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2017.02.039
Keywords
Dissipative Kirchhoff systems; Time-dependent nonlinear damping forces; Local and global asymptotic stability; Fractional p-Laplacian operator
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Funding
- Italian MIUR Project Variational methods, with applications to problems in mathematical physics and geometry [2015KB9WPT_009]
- INdAM - GNAMPA Project Problemi variazionali su varieta Riemanniane e gruppi di Carnot [Prot_2016_000421]
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This paper is devoted to the question of global and local asymptotic stability for nonlinear damped Kirchhoff systems, with homogeneous Dirichlet boundary conditions, under fairly natural assumptions on the external force f = f (t, x, u), the distributed damping Q = Q(t, x, u, u(t)), the perturbation term mu|u|(p-2)u and the dissipative term rho(t)M([u](s)(p))|u(t)|(P-2)u(t), with rho >= 0 and in L-loc(1) (R-0(+)), when the initial data are in a special region. Here u = (u1,..., u(N)) = u(t, x) represents the vectorial displacement, with N >= 1. Particular attention is devoted to the asymptotic behavior of the solutions in the linear case specified in Section 5. Finally, the results are extended to problems where the fractional p-Laplacian is replaced by a more general elliptic nonlocal integro-differential operator. The paper extends in several directions recent theorems and covers also the so-called degenerate case, that is the case in which M is zero at zero. (C) 2017 Elsevier Inc. All rights reserved.
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