The beam equation with nonlinear memory

In this paper, we study the critical exponent for the beam equation with nonlinear memory, i.e., u(tt) + Delta(2)u = F(t, u), where F = integral(t)(0) f(t - s)N(u)(s, x) ds, N(u) approximate to vertical bar u vertical bar(p). For suitable f and p, we prove the existence of local-in-time solutions and small data global solutions to the Cauchy problem, in homogeneous and nonhomogeneous Sobolev spaces. In some cases, we prove that the local solution cannot be extended to a global one. We also consider the limit case of power nonlinearity, i.e., F = N(u).