The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms

This paper deals with the critical exponents for the quasi-linear parabolic equations in R-n and with an inhomogeneous source, or in exterior domains and with inhomogeneous boundary conditions. For n >= 3, sigma > -2/n and p > max {1, 1 + sigma}, we obtain that p(c) = n(1 + sigma)/(n - 2) is the critical exponent of these equations. Furthermore, we prove that if max {1, 1 + sigma} < p <= p(c) then every positive solution of these equations blows up in finite time; whereas these equations admit the global positive solutions for some f(x) and some initial data u(0)(x) if p > p(c) Meantime, we also demonstrate that every positive solution of these equations blows up in finite time provided n = 1, 2, sigma > -1 and p > max {1, 1 + sigma}. (C) 2006 Elsevier Inc. All rights reserved.