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 typeut=−(−Δ)mu−|u|p−1uinRN×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.