We investigate the large-time behavior of classical solutions to the thin-film type equation u t = −(uu xxx ) x . It was shown in previous work of Carrillo and Toscani that for non-negative initial data u 0 that belongs to H 1 (R) and also has a finite mass and second moment, the strong solutions relax in the L 1 (R) norm at an explicit rate to the unique self-similar source type solution with the same mass. The equation itself is gradient flow for an energy functional that controls the H 1 (R) norm, and so it is natural to expect that one should also have convergence in this norm. Carrillo and Toscani raised this question, but their methods, using a different Lyapunov functions that arises in the theory of the porous medium equation, do not directly address this since their Lyapunov functional does not involve derivatives of u.Here we show that the solutions do indeed converge in the H 1 (R) norm at an explicit, but slow, rate. The key to establishing this convergence is an asymptotic equipartition of the excess energy. Roughly speaking, the energy functional whose dissipation drives the evolution through gradient flow consists of two parts: one involving derivatives of u, and one that does not. We show that these must decay at related rates-due to the asymptotic equipartition-and then use the results of Carrillo and Toscani to control the rate for the part that does not depend on derivatives. From this, one gets a rate on the dissipation for all of the excess energy.

Abstract. We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems likewith vanishing conditions at infinity. Here µ is a bounded Radon measure whose support is compactly contained in R N , N ≥ 2, and −(∆) s is the fractional Laplace operator of order s ∈ (1/2, 1).

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We prove a lower bound on the rate of relaxation to equilibrium in the H 1 norm for a thin film equation. We find a two stage relaxation, with power law decay in an initial interval, followed by exponential decay, at an essentially optimal rate, for large times. The waiting time until the exponential decay sets in is explicitly estimated.

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