Pointwise Bounds for the Solutions of the Smoluchowski Equation with Diffusion

We prove various decay bounds on solutions (f _n : n > 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n ^? f _n in terms of a suitable average of the moments of the initial data for every positive ?. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of ${L^p(\mathbb{R}^d \times 0, T])}$ norms of the moments ${X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}$ , ( ${\int_0^{\infty} m^a f_m(x, t)dm}$ in the case of continuous Smoluchowski’s equation) for every ${p \in 1, \infty]}$ . In previous papers 11] and 5] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient d(n) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function ${\phi(n)}$ that is closely related to the total increase of the diffusion coefficient in the interval (0, n].