Remarks on a nonlocal problem involving the Dirichlet energy

By Michel Chipot, V. Valente, G. V. Caffarelli. 2003

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Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms (2005, unknown): “(3.3) Since ‖∇u‖2 = ∫ Ω −u ·∆udx ≤ ‖u‖‖∆u‖ ≤ Cs‖∇u‖‖∆u‖, (3.4) i.e., ‖∇u‖ ≤ Cs‖∆u‖, (3.5) we deduce from (3.3) that ‖∇u‖2 ≤ e− α C2s t‖∇u0‖2 + C2s‖f‖2 α2 , (3.6) and the second estimate in (3.1) follows. the first estimate in (3.1) directly follows from the Poincaré inequality.”
Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms (2015, background): “As said before, a kind of Lyapunov structure were available only in very special situations (see [12, 14, 15]).”