A Liouville comparison principle for sub- and super-solutions of the equation w_t-Δ_p (w) = {pipe}w{pipe}~(q-1)w

摘要

We establish a Liouville comparison principle for entire weak sub- and super-solutions of the equation (*) w_t-Δ_p(w) = {pipe}w{pipe}~(q-1)w in the half-space S = ?_1~+ × ?~n, where n ≥ 1, q > 0, and -Δ_p (w):= div_x({pipe}Δ_xw{pipe}~(p-2)Δ_xw), 1 < p ≤ 2. In our study we impose neither restrictions on the behaviour of entire weak sub- and super-solutions of (*) on the hyper-plane t = 0, nor any growth conditions on their behaviour and on that of any of their partial derivatives at infinity. We prove that if 1 < q < p - 1 + p and u and v are, respectively, an entire weak super-solution and an entire weak sub-solution of (*) in S which belong, only locally in S, to the corresponding Sobolev space and are such that u ≥ v, then u ≡ v. The result is sharp. As direct corollaries we obtain known Fujita-type and Liouville-type theorems.