Let $E=\{u\in C^1[0,1]:u\left(0\right)=u\left(1\right)=0\}$. Let $S_k^{\nu }$ with $\nu =\{+,-\}$ denote the set of functions $u\in E$ which have exactly $k-1$ interior nodal zeros in (0, 1) and $\nu u$ be positive near $0$. We show the existence of $S$-shaped connected component of $S_k^{\nu }$-solutions of the problem $$\left\{\begin{array}{cc}{\left({\displaystyle \frac{u^{\text{'}}}{\sqrt{1-u^{\text{'}2}}}}\right)}^{\text{'}}+\lambda a\left(x\right)f\left(u\right)=0,\hfill & x\in (0,1),\hfill \\ u\left(0\right)=u\left(1\right)=0,\hfill \end{array}\right.$$ where $\lambda >0$ is a parameter, $a\in C\left(\right[0,1],(0,\infty \left)\right)$. We determine the intervals of parameter $\lambda$ in which the above problem has one, two or three $S_k^{\nu }$-solutions. The proofs of the main results are based upon the bifurcation technique.

We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative...

In this note, we study the scattering amplitude for the Schrödinger equation with constant magnetic field. We consider the case where the strengh of the magnetic field goes to infinity and we discuss the competition between the magnetic and the electrostatic effects.

We study the decay in time of solutions of a symmetric regularized-long-wave equation and we show that under some restriction on the form of nonlinearity, the solutions of the nonlinear equation have the same long time behavior as those of the linear equation. This behavior allows us to establish a nonlinear scattering result for small perturbations.

We consider scattering properties of the critical nonlinear system of wave equations with Hamilton structure ⎧uₜₜ - Δu = -F₁(|u|²,|v|²)u, ⎨ ⎩vₜₜ - Δv = -F₂(|u|²,|v|²)v, for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). By using the energy-conservation law over the exterior of a truncated forward light cone and a dilation identity, we get a decay estimate for the potential...

Two approaches are proposed to modelling of topological variations in elastic solids. The first approach is based on the theory of selfadjoint extensions of differential operators. In the second approach function spaces with separated asymptotics and point asymptotic conditions are introduced, and a variational formulation is established. For both approaches, accuracy estimates are derived.

Large time behavior of solutions to the generalized damped wave equation $u_{tt}+A{u}_t+\nu Bu+F(x,t,u,u_t,\nabla u)=0$ for $(x,t)\in {ℝ}^n\times [0,\infty )$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $A{u}_t=u_t$, $Bu=-\Delta u$, and the nonlinear term is either $|u_t{|}^{q-1}{u}_t$ or ${\left|u\right|}^{\alpha -1}u$. In this case, the asymptotic profile of solutions is given...

We investigate the two-component Nernst-Planck-Debye system by a numerical study of self-similar solutions using the Runge-Kutta method of order four and comparing the results obtained with the solutions of a one-component system. Properties of the solutions indicated by numerical simulations are proved and an existence result is established based on comparison arguments for singular ordinary differential equations.

In this paper we examine self-similar solutions to the system $u_{it}-d_i\Delta u_i={\prod }_{k=1}^m{u}_k^{p_k^i}$, i = 1,…,m, $x\in {ℝ}^N$, t > 0, $u_i(0,x)=u_{0i}\left(x\right)$, i = 1,…,m, $x\in {ℝ}^N$, where m > 1 and $p_k^i>0$, to describe asymptotics near the blow up point.

We consider a system which describes the scaling limit of several chemotaxis systems. We focus on self-similarity, and review some recent results on forward and backward self-similar solutions to the system.

In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying...

We consider a sequence of multi-bubble solutions $u_k$ of the following fourth order equation$${\Delta }^2{u}_k={\rho }_k\frac{h\left(x\right)e^{u_k}}{{\int }_{\Omega }h{e}^{u_k}}\text{in}\Omega ,u_k=\Delta u_k=0\text{on}\partial \Omega ,(*)$$where $h$ is a $C^{2,\beta }$ positive function, $\Omega$ is a bounded and smooth domain in ${ℝ}^4$, and ${\rho }_k$ is a constant such that ${\rho }_k\le C$. We show that (after extracting a subsequence), ${lim}_{k\to +\infty }{\rho }_k=32{\sigma }_{3}m$ for some positive integer $m\ge 1$, where ${\sigma }_3$ is the area of the unit sphere in ${ℝ}^4$. Furthermore, we obtain the following sharp estimates for ${\rho }_k$:$$\begin{array}{cc}\hfill {\rho }_k-32{\sigma }_{3}m& =c_0\sum _{j=1}^m{ϵ}_{k,j}^2\left(\sum _{l\ne j}\Delta G_4(p_j,p_l)+\Delta R_4(p_j,p_j)+\frac{1}{32{\sigma }_3}\Delta logh\left(p_j\right)\right)\hfill \\ & +o\left(\sum _{j=1}^m{ϵ}_{k,j}^2\right)\hfill \end{array}$$where $c_0\>0$, $log\frac{64}{{ϵ}_{k,j}^4}=\underset{x\in B_{\delta }\left(p_j\right)}{max}{u}_k\left(x\right)-log\left(\underset{\Omega }{\int }h{e}^{u_k}\right)$ and $u_k\to 32{\sigma }_3\sum _{j=1}^m{G}_4(·,p_j)$ in $C_{\mathrm{loc}}^4(\Omega \setminus \{p_1,...,p_m\})$. This yields a bound of solutions as ${\rho }_k$ converges to $32{\sigma }_{3}m$ from below provided that$$\sum _{j=1}^m\left(\sum _{l\ne j}\Delta G_4(p_j,p_l)+\Delta R_4(p_j,p_j)+\frac{1}{32{\sigma }_3}\Delta logh\left(p_j\right)\right)\>0.$$The analytic work of...