The Kirchhoff Equation for the P – Laplacian

wt t (t, x)− K (‖wx (t, ·)‖ β Lr (R))a(wx (t, x))wxx (t, x) = 0, (2) w(0, x) = 8(x), wt (0, x) = 9(x), where K is an arbitrary function, sufficiently smooth and taking only positive values; and a = a(s) behaves like |s|p−2 near s = 0. The detailed assumptions on K , r , β, and a are given in (3), (4) and Condition 1 below. For K = K (s) = c1 + c2s (c1, c2 > 0) and p = r = β = 2, we get the famous Kirchhoff equation, proposed by Kirchhoff [11] for a better description of the motion of a stretched string. The global existence for real analytic initial data was proved in [1] and [14], while the global existence of small C and Sobolev solutions was established in [3] and [6]. The question of global solutions for arbitrary data from Sobolev spaces is still open. The situation becomes even more delicate if we replace the Laplacian by a nonlinear differential operator: suppose K = K (s) ≡ 1, and consider the equation