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

Abstract

This paper deals with the Cauchy problem of inhomogeneous evolution P-Laplacian equations ∂ t u − div ( | ∇ u | p − 2 ∇ u ) = u q + w ( x ) with nonnegative initial data, where p > 1 , q > max { 1 , p − 1 } , and w ( x ) ⁄ ≡ 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 2 n / ( 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 ( x ) and some initial data. Meanwhile, we also prove that its positive solutions blow up in finite time provided that n ≤ p .