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 Rn and with an inhomogeneous source, or in exterior domains and with inhomogeneous boundary conditions. For n 3, σ >−2/n and p >max{1, 1 + σ}, we obtain that pc = n(1 + σ)/(n − 2) is the critical exponent of these equations. Furthermore, we prove that if max{1, 1 + σ} < p pc, 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 u0(x) if p >pc. Meantime, we also demonstrate that every positive solution of these equations blows up in finite time provided n = 1, 2, σ >−1 and p >max{1, 1+σ}. © 2006 Elsevier Inc. All rights reserved.