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

Abstract

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 , σ > − 2 / n and p > max { 1 , 1 + σ } , we obtain that p c = n ( 1 + σ ) / ( n − 2 ) is the critical exponent of these equations. Furthermore, we prove that if max { 1 , 1 + σ } < 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 , σ > − 1 and p > max { 1 , 1 + σ } .