L ∞ and decay estimates for higher-order semilinear diffusion–absorption equations

We derive estimates of solutions of the semilinear 2mth-order parabolic equation of diffusion–absorption type ut =−(− )mu − |u|p−1u in RN ×R+, m 2, p>1, with bounded initial data u0 from Lq or other functional spaces. For m = 1, i.e., for the semilinear heat equation with absorption intensively studied from the 1970s, basic global L ∞-estimates are straightforward and guaranteed by the Maximum Principle. We show that for m 2, where comparison or order-preserving properties of parabolic flows fail, some similar estimates can be obtained by scaling techniques establishing the rates of decay of the solutions as t→∞and the behaviour as t →0. © 2007 Elsevier Inc. All rights reserved.