This work proposes a description of the set of symmetric vortices, defined as specific solutions of the Ginzburg-Landau equations for a superconducting cylinder with applied magnetic field. It is conducted through a two parameters shooting procedure which relates the behaviour of a symmetric vortex at the center to its behaviour at the boundary. The main result is that, for a given degree $d$, the set of parameters for which such a "shooting" leads to a "response" -- i.e., admissible values for the radius $\bar r$ of the cylinder and the intensity $h$ of the magnetic field -- is a bounded subset in $\mathbb R^2$. This shows in particular that, for large intensities of the applied magnetic field, normal states do not appear as a limit of superconducting vortices of given degree, and that symmetric vortices are not equilibrium states of the system for too large or too low intensities of the applied magnetic field. Moreover, a simpler proof for the existence of bifurcations (a model for phase transitions) from the normal state to superconducting states, as studied in [11], is provided.

We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with a harmonic potential $i\phi_t+\frac{1}{2}\bigtriangleup\phi- \frac{1}{2}\omega^2|x|^2\phi+|\phi|^{4/N}\phi=0,\quad x \in R^N,\quad t \geq 0,$ which models the Bose-Einstein condensate. We establish the lower bound of blow-up rate as $t\rightarrow T$. Furthermore, the $L^2-$concentration property of the radially symmetric blow-up solutions is obtained.

We consider the nonlinear Sturm-Liouville problem $$ -u''(t) + f(u(t)) = \lambda u(t), \ \ u(t) > 0, \quad t \in I := (0, 1), \ \ u(0) = u(1) = 0, $$ where $\lambda > 0$ is an eigenvalue parameter. For better understanding of the global behavior of the branch of positive solutions in $\mbox{\bf R}_+ \times L^q(I)$ ($1 \le q \le \infty$), we establish precise asymptotic formulas for the eigenvalue $\lambda$ with respect to $\Vert u_\lambda\Vert_q$, where $u_\lambda$ is the unique solution associated with given $\lambda > \pi^2$.

We consider the dynamic elasticity equations with a locally distributed damping in a bounded domain. The local dissipation of the form $a(x)y_{t}$ allows coefficients $a$ that lie in some $L^r(\Omega)$, with $(r>2)$. Using multiplier techniques, interpolation inequalities, and a judicious application of the Hölder inequality, we prove sharp energy decay estimates for all $r>Max(2,N)$, where $N$ denotes the space dimension. All space dimensions are considered; the results obtained generalize and improve earlier works where $r$ is required to satisfy $r \! \geq \! {3N+\sqrt{9N^2-16N}\over4}$, for $N\geq3$.

We study the Cauchy problem for the semilinear wave equation in the Schwarzschild metric ($(3+1)$--dimensional space--time). First, we establish that the problem is locally well posed in $ \mathrm H^\sigma$ for any $\sigma \in [1,p+1)$; then we prove the blow--up of the solution in two cases: a)} $p \in (1,1+\sqrt{2})$ and small initial data supported far away from the black hole, b) $p \in (2,1+\sqrt{2})$ and large data supported near the black hole. In both cases, we also give an estimate from above for the lifespan of the solution.

A system of ordinary differential equations $dx/dt=X(t,x)$ which has a zero solution $x=0$ is considered. It is assumed that there exists a function $V(t,x)$, positive definite with respect to part of state variables such that its derivative $dV/dt$ is nonpositive. It is proved that if the function $\sum_{i=1}^jV_i^2$ is positive definite with respect to part of the studying variables, then the zero solution is asymptotically stable with respect to these variables. Here $V_1=dV/dt, V_{i}=dV_{i-1}/dt, \quad i=2, \dots,j;\quad j$ is some positive integer. The instability criterion is also obtained.