We show that in a smooth bounded domain Ω ⊂ ℝn, n ≥ 2, all global nonnegative solutions of ut- Δum= upwith zero boundary data are uniformly bounded in Ω × (τ,∞) by a constant depending on Ω, p; and τ but not on u0, provided that 1 < m < p < [(n + 1)/(n - 1)]m. Furthermore, we prove an a priori bound in L∞(Ω × (0,∞)) depending on ||u0||L∞(Ω)under the optimal condition 1 < m < p < [(n + 2)/(n - 2)]m. © 2004 Elsevier Ltd. All rights reserved.
Winkler, M. (2004). Universal bounds for global solutions of a forced porous medium equation. Nonlinear Analysis, Theory, Methods and Applications, 57(3), 349–362. https://doi.org/10.1016/j.na.2004.02.019