We study the energy transfer in the linear system $$ begin{cases} ddot u+u+dot u=bdot v ddot v+v-epsilon dot v=-bdot u end{cases} $$ made by two coupled differential equations, the first one dissipative and the second one antidissipative. We see how the competition between the damping and the antidamping mechanisms affect the whole system, depending on the coupling parameter $b$.
We prove that a family of linear bounded evolution operators $({bf G}(t,s))_{tge sin I}$ can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators $bm{mathcal A}$ with unbounded coefficients defined in $Itimes Rd$ (where $I$ is a right-halfline or $I=R$) all having the same principal part. We establish some continuity and representation properties of $({bf G}(t,s))_{t ge sin I}$ and a sufficient condition for the evolution operator to be compact in $C_b(Rd;R^m)$. We prove also a uniform weighted gradient estimate and some of its more relevant consequence.
We obtain multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals. There is no assumption to the effect that the nonlinear terms have to satisfy the Ambrosetti-Rabinowitz superquadratic condition. To establish the existence of solutions, a new version of the symmetric mountain pass theorem for strongly indefinite functionals is presented in this paper. This theorem is subsequently applied to deal with cases where all the Palais-Smale sequences of the energy functional may be unbounded.
This note is devoted to the study of Hyt{o}nens extrapolation theorem of compactness on weighted Lebesgue spaces. Two criteria of compactness of linear operators in the two-weight setting are obtained. As applications, we obtain two-weight compactness of commutators of Calder{o}n--Zygmund operators, fractional integrals and bilinear Calder{o}n--Zygmund operators.
Let $f$ be a transcendental meromorphic function, defined in the complex plane $mathbb{C}$. In this paper, we give a quantitative estimations of the characteristic function $T(r,f)$ in terms of the counting function of a homogeneous differential polynomial generated by $f$. Our result improves and generalizes some recent results.