In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form u t (x, t) = R d G(x − y)(u(y, t) − u(x, t)) dy. For example, we will consider equations like,and a nonlocal analogous to the p-Laplacian,p−2 u(y, t) − u(x, t) dy.The energy method developed here allows us to obtain decay rates of the form,for some explicit exponent α that depends on the parameters, d, q and p, according to the problem under consideration.

We consider semidiscrete approximation schemes for the linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semidiscretization scheme we show that, as the mesh size tends to zero, the semidiscrete approximate solutions lose the dispersion property. This fact is proved by constructing solutions concentrated at the points of the spectrum where the second order derivatives of the symbol of the discrete Laplacian vanish. Therefore this phenomenon is due to the presence of numerical spurious high frequencies. To recover the dispersive properties of the solutions at the discrete level, we introduce two numerical remedies: Fourier filtering and a two-grid preconditioner. For each of them we prove Strichartz-like estimates and a local space smoothing effect, uniform in the mesh size. The methods we employ are based on classical estimates for oscillatory integrals. These estimates allow us to treat nonlinear problems with L 2-initial data, without additional regularity hypotheses. We prove the convergence of the two-grid method for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates.

In this paper we study a nonlocal equation that takes into account convective and diffusive effects,with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convection-diffusion equation u t = u + b · ∇(f (u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convection-diffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t → ∞ when f (u) = |u| q−1 u with q > 1. We find the decay rate and the first-order term in the asymptotic regime.

We introduce a splitting method for the semilinear Schrödinger equation and prove its convergence for those nonlinearities which can be handled by the classical well-posedness L 2 (R d )-theory.More precisely, we prove that the scheme is of first order in the L 2 (R d )-norm for H 2 (R d )-initial data.