Existence of positive periodic solutions to nonlinear second order differential equations

In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation $u^{″}\left(t\right)+a\left(t\right)u\left(t\right)=f(t,u\left(t\right))\text{,}t\in R\text{,}$ where $a:R\to [0,+\infty )$ is an $\omega$-periodic continuous function with $a\left(t\right)⁄\equiv 0$, $f:R\times [0,+\infty )\to [0,+\infty )$ is continuous and $f(\cdot ,u):R\to [0,+\infty )$ is also an $\omega$-periodic function for each $u\in [0,+\infty )$. Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.     

Asymptotics for some semilinear hyperbolic equations with non-autonomous damping

Let V and H be Hilbert spaces such that $V?H?V^{\prime }$ with dense and continuous injections. Consider a linear continuous operator $A:V\to V^{\prime }$ which is assumed to be symmetric, monotone and semi-coercive. Given a function $f:V\to H$ and a map $\gamma \in W_{\mathit{loc}}^{1,1}({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ such that ${\lim }_{t\to +\infty }\gamma (t)=0$, our purpose is to study the asymptotic behavior of the following semilinear hyperbolic equation$\frac{d^{2}u}{d{t}^2}(t)+\gamma (t)\frac{du}{dt}(t)+Au(t)+f(u(t))=0,t?0.$ The nonlinearity f is assumed to be monotone and conservative. Condition ${\int }_0^{+\infty }\gamma (t)dt=+\infty$ guarantees that some suitable energy function tends toward its minimum. The main contribution of this paper is to provide a general result of convergence for the trajectories of (E): if the quantity $\gamma (t)$ behaves as $k/t^{\alpha }$, for some $\alpha \in ]0,1[$, $k>0$ and t large enough, then $u(t)$ weakly converges in V toward an equilibrium as $t\to +\infty$. Strong convergence in V holds true under compactness or symmetry conditions. We also give estimates for the speed of convergence of the energy under some ellipticity-like conditions. The abstract results are applied to particular semilinear evolution problems at the end of the paper.     

The generic gradient-like structure of certain asymptotically autonomous semilinear parabolic equations

We consider asymptotically autonomous semilinear parabolic equations

$u_t+Au=f(t,u).$

Suppose that $f(t,.)\to f^{\pm }$ as $t\to \pm \infty$, where the semiflows induced by

(*)$u_t+Au=f^{\pm }(u)$

are gradient-like. Under certain assumptions, it is shown that generically with respect to a perturbation g with $g(t)\to 0$ as $\left|t\right|\to \infty$, every solution of

$u_t+Au=f(t,u)+g(t)$

is a connection between equilibria $e^{\pm }$ of (*) with $m(e^{?})\ge m(e^{+})$. Moreover, if the Morse indices satisfy $m(e^{?})=m(e^{+})$, then u is isolated by linearization.     

Traveling waves in a convolution model with infinite distributed delay and non-monotonicity

In this paper, we study a population model with nonlocal diffusion and a non-monotonic reaction term with infinite distributed delay. Some existence results of traveling wavefronts for the system with a monotonic reaction term are firstly obtained by the construction of upper-lower solutions and the application of Schauder’s fixed point theorem. Then by constructing a couple of auxiliary equations with monotonicity and using the comparison method, we prove the existence of traveling waves for the system without monotonicity. We also give some discussion on the asymptotic behavior of the traveling waves as $\xi =x+ct\to ?\infty$.     

Semi-global weak stabilization of bilinear Schrödinger equations

We consider a linear Schrödinger equation, on a bounded domain, with bilinear control, representing a quantum particle in an electric field (the control). Recently, Nersesyan proposed explicit feedback laws and proved the existence of a sequence of times ${(t_n)}_{n\in \mathbb{N}}$ for which the values of the solution of the closed loop system converge weakly in $H^2$ to the ground state. Here, we prove the convergence of the whole solution, as $t\to +\infty$. The proof relies on control Lyapunov functions and an adaptation of the LaSalle invariance principle to PDEs.