A class of dissipative nonautonomous nonlocal second-order evolution equations

In this paper we consider the following nonlinear and spatially nonlocal second-order evolution equation from nonlocal theory of continuum mechanics {u(tt) + a(t, x) u(t) - integral(RN) J(x, y)(beta u(y) - u(x)) dy = f (u), x is an element of Omega, t > tau, u(x, tau) = u(0)(x), u(tau) (x, tau) = u(1)(x), x is an element of Omega, u(x, t) = 0, x is an element of R-n\Omega, t >= tau, where Omega is a bounded smooth domain in R-n, n >= 3, 0 < beta < 1, and a is a bounded continuous function. Here, the kernel J is a nonnegative, symmetric bounded function with bounded derivative, satisfying certain growth conditions. We deduce an energy functional associated to these problem, and we study the local and global well posedness, boundedness and asymptotic behavior of its solutions. Additionally we study the stability of the trivial solution associated to these problem.