Asymptotic stability for nonlinear damped Kirchhoff systems involving the fractional p-Laplacian operator

Abstract

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,ut), the perturbation term μ|u|p−2u and the dissipative term ϱ(t)M([u]sp)|ut|p−2ut, with ϱ≥0 and in Lloc1(R0+), when the initial data are in a special region. Here u=(u1,…,uN)=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.