An optimal Liouville-type theorem of the quasilinear parabolic equation with a pp-Laplace operator

In this paper, we consider nonnegative solutions of the quasilinear parabolic equation with pp-Laplace operator View the MathML sourceut=div(|∇u|p−2∇u)+|u|q−1u, where p>2p>2 and q>p−1q>p−1. Our main result is that there is no nontrivial positive bounded radial entire solution. The proof is based on intersection comparison arguments, which can be viewed as a sophisticated form of the maximum principle and has been used to deal with the semilinear heat equation by Poláčik and Quittner [Peter Poláčik, Pavol Quittner, A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation, Nonlinear Analysis TMA 64 (2006) 1679–1689] and the porous medium equation by Souplet [Ph. Souplet, An optimal Liouville-type theorem for radial entire solutions of the porous medium equation with source, J. Differential Equations 246 (2009) 3980–4005].