We also exhibit a set of localized data for which the corresponding solutions are strongly focused, which in geometric terms means that a wave travels along an specific incoming null geodesic in such a way that almost all of the energy is concentrated in a tubular neighborhood of the geodesic and almost no energy radiates out of this neighborhood.

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $$\left\{\begin{array}{c}-{\mathrm{div}\left(\right|x|}^{-\alpha p}{\left|\nabla u\right|}^{p-2}{\nabla u)=|x|}^{-(\alpha +1)p+\beta }\left(a{u}^{p-1}-f\left(u\right)-{\displaystyle \frac{c}{u^{\gamma }}}\right),x\in \Omega ,\hfill \\ u=0,x\in \partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a bounded smooth domain of ${ℝ}^N$ with $0\in \Omega$, $1<p<N$, $0\le \alpha <(N-p)/p$, $\gamma \in (0,1)$, and $a$, $\beta$, $c$ and $\lambda$ are positive parameters. Here $f:[0,\infty )\to ℝ$ is a continuous function. This model arises in the studies of population biology of one species with $u$ representing the concentration of the species. We discuss the existence of a positive solution when $f$ satisfies certain additional conditions. We use the method of sub-supersolutions...

This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $-\Delta u\left(x\right)+\left(\omega +\frac{h^2\left(\right|x\left|\right)}{{\left|x\right|}^2}+{\int }_{\left|x\right|}^{+\infty }\frac{h\left(s\right)}{s}{u}^2\left(s\right)ds\right)u\left(x\right)={\left|u\left(x\right)\right|}^{p-1}u\left(x\right)$, where $h\left(r\right)=\frac{1}{2}{\int }_0^{r}s{u}^2\left(s\right)ds$. This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in (1,3)$, the functional may be bounded from below or not, depending on $\omega$. Quite surprisingly, the threshold value for $\omega$ is explicit. From...

The main result of this talk is a global existence theorem for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds.The proof is based on a bootstrap argument involving $L^2$ and $L^{\infty }$ estimates. The $L^2$ bounds are proved in the paper [5]. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation...

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered. A new analytical method for solving an initial value problem for this system is the main object of the paper. This method consists in the following: the initial value problem is written in terms of Fourier images with respect to lateral space variables, then the resulting problem...

This note presents an elementary approach to the nonexistence of solutions of linear parabolic initial-boundary value problems considered in the Feller test.

This paper is concerned with the nonlinear theory of equilibrium for materials which do not conduct electricity. An existence and uniqueness result is established.

We discuss the effect of time delay on blow-up of solutions to initial-boundary value problems for nonlinear reaction-diffusion equations. Firstly, two examples are given, which indicate that the delay can both induce and prevent the blow-up of solutions. Then we show that adding a new term with delay may not change the blow-up character of solutions.

This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function $\chi \left(v\right)$ and the growth term $f\left(u\right)$ under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that $0<\chi \left(v\right)\le {\chi }_0/v^k$$(k\ge 1...$

This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation $${\displaystyle \frac{\partial u}{\partial t}}-(\lambda +\mathrm{i}\alpha )\Delta u+(\kappa +\mathrm{i}\beta ){\left|u\right|}^{q-1}u-\gamma u=0$$ in ${ℝ}^N\times (0,\infty )$ with $L^p$-initial data $u_0$ in the subcritical case ($1\le q<1+2p/N$), where $u$ is a complex-valued unknown function, $\alpha$, $\beta$, $\gamma$, $\kappa \in ℝ$, $\lambda >0$, $p>1$, $\mathrm{i}=\sqrt{-1}$ and $N\in \mathbb{N}$. The proof is based on the $L^p$-$L^q$ estimates of the linear semigroup $\{exp(t(\lambda +\mathrm{i}\alpha )\Delta \left)\right\}$ and usual fixed-point argument.

Nonlinear Schrödinger equations (NLS)${}_a$ with strongly singular potential ${a\left|x\right|}^{-2}$ on a bounded domain $\Omega$ are considered. If $\Omega ={ℝ}^N$ and $a>-{(N-2)}^2/4$, then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a=-{(N-2)}^2/4$ is excluded because $D\left(P_{a\left(N\right)}^{1/2}\right)$ is not equal to $H^1\left({ℝ}^N\right)$, where $P_{a\left(N\right)}:=-\Delta -{(N-2)}^2/{\left(4\right|x|}^2)$ is nonnegative and selfadjoint in $L^2\left({ℝ}^N\right)$. On the other hand, if $\Omega$ is a smooth and bounded domain with $0\in \Omega$, the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000)....

This paper addresses analytical investigations of degenerating PDE systems for phase separation and damage processes considered on nonsmooth time-dependent domains with mixed boundary conditions for the displacement field. The evolution of the system is described by a degenerating Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a quasi-static balance equation for the displacement field. The analysis is performed on a time-dependent...

This paper discusses the existence theory to dynamical model of viscoplasticity and show possibility to obtain existence of solution without assuming weak safe-load condition.

In this paper, we review several recent results dealing with elliptic equations with non local diffusion. More precisely, we investigate several problems involving the fractional laplacian. Finally, we present a conformally covariant operator and the associated singular and regular Yamabe problem.

Nous rappelons tout d’abord l’approche maintenant classique de renormalisation pour établir l’unicité des solutions faibles des équations de transport linéaires, en mentionnant les résultats récents qui s’y rattachent. Ensuite, nous montrons comment l’approche alternative introduite par Crippa et DeLellis estimant directement le flot lagrangien permet d’obtenir des résultats nouveaux. Nous établissons l’existence et l’unicité du flot associé à une équation de transport dont le coefficient a un gradient...