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

This paper deals with the Cauchy problem of inhomogeneous evolution P-Laplacian equations ∂tu−div(|∇u|p−2∇u)=uq+w(x) with nonnegative initial data, where p>1,q>max{1,p−1}p>1,q>max{1,p−1}, and w(x)⁄≡0w(x)⁄≡0 is a nonnegative continuous functions in Rn. We prove that qc=(p−1)n/(n−p)qc=(p−1)n/(n−p) is its critical exponent provided that 2n/(n+1)qcq>qc, the equation possesses a global positive solution for some w(x)w(x) and some initial data. Meanwhile, we also prove that its positive solutions blow up in finite time provided that n≤pn≤p.