Blow-up results and global existence of positive solutions for the inhomogeneous evolution P-Laplacian equations

This paper deals with the Cauchy problem of inhomogeneous evolution P-Laplacian equations at partial derivative(t)u-div(vertical bar del u vertical bar(p-2)del u) = u(q) + w(u) with nonnegative initial data, where p > 1, q > maxf {1, p - 1}, and w(x) not equivalent to 0 is a nonnegative continuous functions in R-n. We prove that q(c) = (p - 1)n/(n - p) is its critical exponent provided that 2n/(n + 1) < p < n, i.e., if q <= q(c), then every positive solution blows up in finite time; whereas for q > q(c) the equation possesses a global positive solution for some w(r) and some initial data. Meanwhile, we also prove that its positive solutions blow up in finite time provided that n < p. (c) 2006 Elsevier Ltd. All rights reserved.