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}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}.