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We study local (the heat equation) and nonlocal (convolution type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. For this star shaped limit problem the asymptotic behaviour of the solutions is just given by the solution to the heat equation in a half line with a Neumann boundary condition at $x = 0$ and initial datum $(2M/N)\delta_{x=0}$ where M is the total mass of the initial condition for our original problem and $N$ is the number of edges of infinite length. In addition, we show that solutions to the nonlocal problem converge, when we rescale the kernel, to solutions to the heat equation (the local problem), that is, we find a relaxation limit., Comment: To appear in Annali di Matematica Pura ed Applicata

We study solutions to the evolution equation ut=Δu−u+∑k⩾1qkuk, t>0, in Rd. Here the coefficients qk⩾0 satisfy ∑k⩾1qk=1

In this paper we consider a nonlocal viscous Burgers' equation and study the well-posedness and asymptotic behavior of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers' equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories.

In this paper we consider a subcritical model that involves nonlocal diffusion and a classical convective term. In spite of the nonlocal diffusion, we obtain an Oleinik type estimate similar to the case when the diffusion is local. First we prove that the entropy solution can be obtained by adding a small viscous term and letting . Then, by using uniform Oleinik estimates for the viscous approximation we are able to prove the well-posedness of the entropy solutions with L 1-initial data. Using a scaling argument and hyperbolic estimates given by Oleinik's inequality, we obtain the first term in the asymptotic behavior of the nonnegative solutions. Finally, the large time behavior of changing sign solutions is proved using the classical flux-entropy method and estimates for the nonlocal operator.

In this paper, we study a non-local diffusion problem that involves three different fractional Laplacian operators acting on two domains. Each domain has an associated operator that governs the diffusion on it, and the third operator serves as a coupling mechanism between the two of them. The model proposed is the gradient flow of a non-local energy functional. In the first part of the paper, we provide results about existence of solutions and the conservation of mass. The second part encompasses results about the [Formula: see text] decay of the solutions. The third part is devoted to study, the asymptotic behavior of the solutions of the problem when the two domains are a ball and its complementary. Exterior fractional Sobolev and Nash inequalities of independent interest are also provided in Appendix A.

In this paper we consider the time dependent one-dimensional Schr\"odinger equation with multiple Dirac delta potentials {of different strengths}. We prove that the classical dispersion property holds under some restrictions on the strengths and on the lengths of the finite intervals. The result is obtained in a more general setting of a Laplace operator on a tree with $\delta$-coupling conditions at the vertices. The proof relies on a careful analysis of the properties of the resolvent of the associated Hamiltonian. With respect to the analysis done in \cite{MR2858075} for Kirchhoff conditions, here the resolvent is no longer in the framework of Wiener algebra of almost periodic functions, and its expression is harder to analyze., Comment: 24p, revised and extended version (repulsive strenghts and general connection conditions considered), to appear in Analysis&PDE

We consider a convection-diffusion model with linear fractional diffusion in the sub-critical range. We prove that the large time asymptotic behavior of the solution is given by the unique entropy solution of the convective part of the equation. The proof is based on suitable a-priori estimates, among which proving an Oleinik type inequality plays a key role., Comment: 24p, typos corrected

In this paper we consider a splitting method for the augmented Burgers equation and prove that it is of first order. We also analyze the large-time behavior of the approximated solution by obtaining the first term in the asymptotic expansion. We prove that, when time increases, these solutions behave as the self-similar solutions of the viscous Burgers equation, 24 pages

We introduce a splitting method for the semilinear Schrodinger 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.

We investigate dispersive and Strichartz estimates for the Schr\"{o}dinger time evolution propagator $\mathrm{e}^{-\mathrm{i}tH}$ on a star-shaped metric graph. The linear operator, $H$, taken into consideration is the self-adjoint extension of the Laplacian, subject to a wide class of coupling conditions. The study relies on an explicit spectral representation of the solution in terms of the resolvent kernel which is further analyzed using results from oscillatory integrals. As an application, we obtain the global well-posedness for a class of semilinear Schr\"{o}dinger equations.

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form u t ( x , t ) = J ∗ u ( x , t ) − u ( x , t ) = ∫ R d J ( x − y ) u ( y , t ) d y − u ( x , t ) in the whole R d with an initial condition u ( x , 0 ) = u 0 ( x ) . Under suitable hypotheses on J (involving its Fourier transform) and u 0 , it is proved an expansion of the form ‖ u ( x , t ) − ∑ | α | ⩽ k ( − 1 ) | α | α ! ( ∫ u 0 ( x ) x α d x ) ∂ α K t ‖ L q ( R d ) ⩽ C t − A , where K t is the regular part of the fundamental solution and the exponent A depends on J , q , k and the dimension d . Moreover, we can obtain bounds for the difference between the terms in this expansion and the corresponding ones for the expansion of v t ( x , t ) = − ( − Δ ) s 2 v ( x , t ) . Here we deal with the case 1 ⩽ q ⩽ 2 . The case 2 ⩽ q ⩽ ∞ was treated previously, by other methods, in L.I. Ignat and J.D. Rossi (2008) [11] .

We analyze the problem of boundary observability of the finite-difference space semi-discretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the mesh-size) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski \cite{0763.76042}. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm for computing the boundary controls for the wave equation. The method can be applied in any space dimension, for more general domains and other discretization schemes.

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form u t = J * u – u in the whole $${\mathbb{R}}^d$$ with an initial condition u(x, 0) = u 0(x). Under suitable hypotheses on J (involving its Fourier transform) and u 0, it is proved an expansion of the form $$\left\| {u(u) - \sum\limits_{\left| \alpha \right| \leq k} {\frac{{( - 1)^{\left| \alpha \right|} }}{{\alpha !}}\left( {\int {u_0 (x)x^\alpha\, dx} } \right)\partial ^\alpha K_t } } \right\| _{L^q ({\mathbb{R}}^d )} \leq Ct^{ - A}$$ , where K t is the regular part of the fundamental solution and the exponent A depends on J, q, k and the dimension d. Moreover, we can obtain bounds for the difference between the terms in this expansion and the corresponding ones for the expansion of the evolution given by fractional powers of the Laplacian, $$\nu_t (x, t) = -(-\Delta)^{\frac{s}{2}} \nu (x, t)$$ .

In this paper we analyze the dispersion for one dimensional wave and Schrodinger equations with BV coefficients. In the case of the wave equation we give a complete answer in terms of the variation of the logarithm of the coefficient showing that dispersion occurs if this variation is small enough but it may fail when the variation goes beyond a sharp threshold. For the Schrodinger equation we prove that the dispersion holds under the same smallness assumption on the variation of the coefficient. But, whether dispersion may fail for larger coefficients is unknown for the Schrodinger equation.

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, $$u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t)$$ with t ≥ 0 and $$n \in {\mathbb{Z}}^{d}$$ . We assume that J is nonnegative and verifies $$\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1$$ . We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.

We consider fully discrete schemes for the one-dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrödinger equation with nonlinearities which cannot be treated by energy methods.

We introduce a two-grid finite difference approximation scheme for the free Schrodinger equation. This scheme is shown to converge and to posses appropriate dispersive properties as the mesh-size tends to zero. A careful analysis of the Fourier symbol shows that this occurs because the two-grid algorithm (consisting in projecting slowly oscillating data into a fine grid) acts, to some extent, as a filtering one. We show that this scheme converges also in a class of nonlinear Schrodinger equations whose well-posedness analysis requires the so-called Strichartz estimates. This method provides an alternative to the method introduced by the authors [L.I. Ignat, E. Zuazua, Dispersive properties of a viscous numerical scheme for the Schrodinger equation, C. R. Math. Acad. Sci. Paris 340 (7) (2005) 529–534] using numerical viscosity. To cite this article: L.I. Ignat, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain $L^1$-$L^p$ decay rates. The asymptotic behavior of the solution is obtained by showing that the influence of the convolution term $K*u_{xx}$ is the same as $u_{xx}$ for large times. Then, we propose a semi-discrete numerical scheme that preserves this asymptotic behavior, by introducing two correcting factors in the discretization of the non-local term. Numerical experiments illustrating the accuracy of the results of the paper are also presented., Comment: Accepted for publication in ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we analyze the long time behavior of the solutions of a nonlocal diffusion-convection equation. We give a new compactness criterion in the Lebesgue spaces $L^p((0,T)\times \Omega)$ and use it to obtain the first term in the asymptotic expansion of the solutions. Previous results of [J. Bourgain, H. Brezis, and P. Mironescu, in Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001] are used to obtain a compactness result in the spirit of the Aubin--Lions--Simon lemma.

We obtain the asymptotic expansion of the solutions of some anisotropic heat equations when the initial data belong to polynomially weighted L p -spaces. We mainly address two model examples. In the first one, the diffusivity is of order two in some variables but higher in the other ones. In the second one, we consider the heat equation on the Heisenberg group.

International audience; In this paper we establish global Carleman estimates for the heat and Schrodinger equations on a network. The heat equation is considered on a general tree and the Schrodinger equation on a star-shaped tree. The Carleman inequalities are used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the heat (resp. the Schrodinger) equation from boundary measurements.

In this paper, we obtain bounds for the decay rate in the Lr (ℝd)-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, ut(x,t)=∫RdK(x,y)|u(y,t)−u(x,t)|p−2(u(y,t)−u(x,t))dy,x∈Rd,t>0. We consider a kernel of the form K(x, y) = ψ(y−a(x)) + ψ(x−a(y)), where ψ is a bounded, nonnegative function supported in the unit ball and a is a linear function a(x)=Ax. To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u)=−∫RdK(x,y)|u(y)−u(x)|p−2(u(y)−u(x))dy,1⩽p

This article is devoted to the analysis of the convergence rates of several numerical approximation schemes for linear and nonlinear Schrodinger equations on the real line. Recently, the authors have introduced viscous and two-grid numerical approximation schemes that mimic at the discrete level the so-called Strichartz dispersive estimates of the continuous Schrodinger equation. This allows to guarantee the convergence of numerical approximations for initial data in L 2 ( R ) , a fact that cannot be proved in the nonlinear setting for standard conservative schemes unless more regularity of the initial data is assumed. In the present article we obtain explicit convergence rates and prove that dispersive schemes fulfilling the Strichartz estimates are better behaved for H s ( R ) data if 0 s 1 / 2 . Indeed, while dispersive schemes ensure a polynomial convergence rate, non-dispersive ones only yield logarithmic ones.

We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T ( u ) = − ∫ R d K ( x , y ) ( u ( y ) − u ( x ) ) d y . Here we consider a kernel K ( x , y ) = ψ ( y − a ( x ) ) + ψ ( x − a ( y ) ) where ψ is a bounded, nonnegative function supported in the unit ball and a means a diffeomorphism on R d . A simple example being a linear function a ( x ) = A x . The upper and lower bounds that we obtain are given in terms of the Jacobian of a and the integral of ψ. Indeed, in the linear case a ( x ) = A x we obtain an explicit expression for the first eigenvalue in the whole R d and it is positive when the determinant of the matrix A is different from one. As an application of our results, we observe that, when the first eigenvalue is positive, there is an exponential decay for the solutions to the associated evolution problem. As a tool to obtain the result, we also study the behavior of the principal eigenvalue of the nonlocal Dirichlet problem in the ball B R and prove that it converges to the first eigenvalue in the whole space as R → ∞ .

In this paper we prove dispersive estimates for the system formed by two coupled discrete Schr\"odinger equations. We obtain estimates for the resolvent of the discrete operator and prove that it satisfies the limiting absorption principle. The decay of the solutions is proved by using classical and some new results on oscillatory integrals., Comment: 27 pages

In this paper, we consider the Schrödinger equation on a network formed by a tree with the last generation of edges formed by infinite strips. We give an explicit description of the solution of the linear Schrödinger equation with constant coefficients. This allows us to prove dispersive estimates, which in turn are useful for solving the nonlinear Schrödinger equation. The proof extends also to the laminar case of positive step-function coefficients having a finite number of discontinuities.

In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut=J∗u−u+G∗(f(u))−f(u) in Rd, 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 ut=Δu+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−1u with q>1. We find the decay rate and the first-order term in the asymptotic regime.

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 ) ) d y . For example, we will consider equations like, u t ( x , t ) = ∫ R d J ( x , y ) ( u ( y , t ) − u ( x , t ) ) d y + f ( u ) ( x , t ) , and a nonlocal analogous to the p-Laplacian, u t ( x , t ) = ∫ R d J ( x , y ) | u ( y , t ) − u ( x , t ) | p − 2 ( u ( y , t ) − u ( x , t ) ) d y . The energy method developed here allows us to obtain decay rates of the form, ‖ u ( ⋅ , t ) ‖ L q ( R d ) ⩽ C t − α , for some explicit exponent α that depends on the parameters, d, q and p, according to the problem under consideration.

In this paper we analyze the large time asymptotic behavior of the discrete solutions of numerical approximation schemes for scalar hyperbolic conservation laws. We consider three monotone conservative schemes that are consistent with the one-sided Lipschitz condition (OSLC): Lax-Friedrichs, Engquist-Osher and Godunov. We mainly focus on the inviscid Burgers equation, for which we know that the large time behavior is of self-similar nature, described by a two-parameter family of N-waves. We prove that, at the numerical level, the large time dynamics depends on the amount of numerical viscosity introduced by the scheme: while Engquist-Osher and Godunov yield the same N-wave asymptotic behavior, the Lax-Friedrichs scheme leads to viscous self-similar profiles, corresponding to the asymptotic behavior of the solutions of the continuous viscous Burgers equation. The same problem is analyzed in the context of self-similar variables that lead to a better numerical performance but to the same dichotomy on the asymptotic behavior: N-waves versus viscous ones. We also give some hints to extend the results to more general fluxes. Some numerical experiments illustrating the accuracy of the results of the paper are also presented., Comment: Error corrected in main theorem in v3. Obtained results do not change in essence. Minor typos corrected in v2