In this paper we study the strict localization for the p-Laplacian equation with strongly nonlinear source term. Let u := u(x, t) be a solution of the Cauchy problem u(t) = div(vertical bar del u vertical bar(p-2)del u) + u(q), u(x,0) = u(0)(x). where (x,t) is an element of R(N) x ((0, T), N >= 1 and p >= 2. When q >= p - 1, we prove that if the initial data u(0)(x) has a compact Support, then the Solution u(., t) has also compactly support. Moreover, when 1 < q < p - 1, we show that the solution of the Cauchy problem blows up at any point of R(N) to arbitrary Compactly Supported initial data. (C) 2008 Elsevier Inc. All rights reserved.