Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space

We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u) = - \int_{\rr^d} K(x,y) (u(y)-u(x)) \, dy. Here we consider a kernel $K(x,y)=\psi (y-a(x))+\psi(x-a(y))$ where $\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ means a diffeomorphism on \rr^d. A simple example being a linear function $a(x)= Ax$. The upper and lower bounds that we obtain are given in terms of the Jacobian of $a$ and the integral of $\psi$. Indeed, in the linear case $a(x) = Ax$ we obtain an explicit expression for the first eigenvalue in the whole \rr^d and it is positive when the 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 behaviour 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 $R\to \infty$

Asymptotic behaviour for fractional diffusion-convection equations

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

Dispersive Properties for Discrete Schrödinger Equations

In this paper we prove dispersive estimates for the system formed by two coupled discrete Schrödinger 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

Decay estimates for nonlinear nonlocal diffusion problems in the whole space

In this paper we obtain bounds for the decay rate in the L^r (\rr^d)-norm for the solutions to a nonlocal and nolinear evolution equation, namely, u_t(x,t) = \int_{\rr^d} K(x,y) |u(y,t)- u(x,t)|^{p-2} (u(y,t)- u(x,t)) \, dy, with x \in \rr^d, $t>0$. Here we consider a kernel $K(x,y)$ of the form $K(x,y)=\psi (y-a(x))+\psi(x-a(y))$, where $\psi$ 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) = - \int_{\rr^d} K(x,y) |u(y)-u(x)|^{p-2} (u(y)-u(x)) \, dy, with $1 \leq p < \infty$. The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole \rr^d: \lambda_{1,p} (\rr^d) = 2(\int_{\rr^d} \psi (z) \, dz)|\frac{1}{|\det{A}|^{1/p}} -1|^p. Moreover, we deal with the $p=\infty$ eigenvalue problem studying the limit as $p \to \infty$ of $\lambda_{1,p}^{1/p}$

Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals

We study a singular nonlinear ordinary differential equation on intervals {[}0, R) with R <= +infinity, motivated by the Ginzburg-Landau models in superconductivity and Landau-de Gennes models in liquid crystals. We prove existence and uniqueness of positive solutions under general assumptions on the nonlinearity. Further uniqueness results for sign-changing solutions are obtained for a physically relevant class of nonlinearities. Moreover, we prove a number of fine qualitative properties of the solution that are important for the study of energetic stability