The aim of this paper is to find estimates of the Green’s function of stationary discrete shock profiles and discrete boundary layers of the modified Lax–Friedrichs numerical scheme, by using techniques developed by Zumbrun and Howard [27] in the continuous viscous setting.

The aim of this paper is to find estimates of the Green's function of stationary discrete shock profiles and discrete boundary layers of the modified Lax–Friedrichs numerical scheme, by using techniques developed by Zumbrun and Howard [CITE] in the continuous viscous setting.

In this preliminary Note we outline the results of the forthcoming paper [2] dealing with a class on nonlinear Schrödinger equations with potentials vanishing at infinity. Working in weighted Sobolev spaces, the existence of a ground state is proved. Furthermore, the behaviour of such a solution, as the Planck constant tends to zero (semiclassical limit), is studied proving that it concentrates at a point.

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V\left(x\right)\sim {\left|x\right|}^{-\alpha }$, $0<\alpha <2$, and $K\left(x\right)\sim {\left|x\right|}^{-\beta }$, $\beta >0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\epsilon }$ belonging to $W^{1,2}\left({ℝ}^N\right)$ is proved under the assumption that $\sigma <p<(N+2)/\left(N-2\right)$ for some $\sigma ={\sigma }_{N,\alpha ,\beta }$. Furthermore, it is shown that $v_{\epsilon }$ are spikes concentrating at a minimum point of $𝒜=V^{\theta }{K}^{-2/\left(p-1\right)}$, where $\theta =(p+1)/\left(p-1\right)-1/2$.

This text aims to describe results of the authors on the long time behavior of NLS on product spaces with a particular emphasis on the existence of solutions with growing higher Sobolev norms.

We study asymptotic behavior of radial solutions of a nonlocal Fokker-Planck equation describing the evolution of self-attracting particles. In particular, we consider stationary solutions in balls and in the whole space, self-similar solutions defined globally in time, blowing up self-similar solutions, and singularities of solutions that blow up in a finite time.

We consider the model, proposed by Dawidowicz and Zalasiński, describing the interactions between the heterotrophic and autotrophic organisms coexisting in a terrestrial environment with available oxygen. We modify this model by assuming intraspecific competition between heterotrophic organisms. Moreover, we introduce a diffusion of both types of organisms and oxygen. The basic properties of the extended model are examined and illustrated by numerical simulations.

We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix $s>1$. Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with $s$-Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is $c>0$ such that for any $𝒦\gg 1$ we find a solution $u$ and a time $T$ such that ${\parallel u\left(T\right)\parallel }_{H^s}\ge 𝒦{\parallel u\left(0\right)\parallel }_{H^s}$. Moreover, the time $T$ satisfies the polynomial bound $0<T<{𝒦}^C$.

We discuss the propagation of electromagnetic waves of a special form through an inhomogeneous isotropic medium which has a cylindrical symmetry and a nonlinear dielectric response. For the case where this response is of self-focusing type the problem is treated in [1]. Here we continue this study by dealing with a defocusing dielectric response. This tends to inhibit the guidance properties of the medium and so guidance can only be expected provided that the cylindrical stratification is such that...

We consider a biharmonic problem ${\Delta }^2{u}_{\omega }=f_{\omega }$ with Navier type boundary conditions $u_{\omega }=\Delta u_{\omega }=0$, on a family of truncated sectors ${\Omega }_{\omega }$ in ${ℝ}^2$ of radius $r$, $0<r<1$ and opening angle $\omega$, $\omega \in (2\pi /3,\pi ]$ when $\omega$ is close to $\pi$. The family of right-hand sides ${\left(f_{\omega }\right)}_{\omega \in (2\pi /3,\pi ]}$ is assumed to depend smoothly on $\omega$ in $L^2\left({\Omega }_{\omega }\right)$. The main result is that $u_{\omega }$ converges to $u_{\pi }$ when $\omega \to \pi$ with respect to the $H^2$-norm. We can also show that the $H^2$-topology is optimal for such a convergence result.

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized....

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy’s version of the uncertainty principle. We also obtain corresponding results for heat evolutions.

In this preliminary Note we outline some results of the forthcoming paper [11], concerning positive solutions of the equation $\partial {}_{t}u=\mathrm{△}u+\frac{c}{\left|x^2\right|}u(0<c<\frac{{\left(n-2\right)}^2}{4};n\ge 3)$. A parabolic Harnack inequality is proved, which in particular implies a sharp two-sided estimate for the associated heat kernel. Our approach relies on the unitary equivalence of the Schrödinger operator $Hu=-\mathrm{△}u-\frac{c}{{\left|x\right|}^2}u$ with the opposite of the weighted Laplacian ${\mathrm{△}}_{\lambda }v=\frac{1}{{\left|x\right|}^{\lambda }}\text{div}\left({\left|x\right|}^{\lambda }\nabla v\right)$ when $\lambda =2-n+2\sqrt{c_0-c}$.