Orlando Lopes

On the spectrum of evolution operators generated by hyperbolic systems

Abstract We prove that the asymptotic behavior of semigroups generated by mixed-initial value problems for hyperbolic systems in one space variable is determined by the zeroes of the characteristic equation. The proof is based on the following reduction principle: we construct a reduced system (which is much simpler than the given one) such that the difference of the semigroups is a compact operator. The construction of the reduced system is presented in the non-autonomous case because it has implifications in the study of systems which are time periodic.

α-contractions and attractors for dissipative semilinear hyperbolic equations and systems

SummaryIn this paper we discuss the existence of compact attractor for the abstract semilinear evolution equation u=Au+f(t, u); the results are applied to damped partial differential equations of hyperbolic type. Our approach is a combination of Liapunov method with the theory of α-eontractions.

Stability and forced oscillations

In this paper we derive a differential-difference equation for a circuit involving a lossless transmission line and we give conditions for global asymptotic stability of an equilibrium point, existence and stability of forced oscillations. Some of such problems have been investigated for an equation obtained by R. K. Brayton [Quart. J. Appl. Math. 24 (1967), 289–301; O. Lopes, SIAM J. Appl. Math., to appear; M. Slemrod, J. Math. Anal. Appl. 36 (1971), 22–40] but, for ours (which governs the same physical problem), better results can be proved. By using suitable Liapunov functionals, we reduce the problem of stability and uniform ultimate boundedness to a scalar ordinary differential inequality.

Stability of solitary waves of some coupled systems

Solitary waves of Hamiltonian dispersive systems arise as critical points of the augmented Lagrangian V(u) + γI1(u) + ωI2(u), where V(u), I1(u) and I2(u) are first integrals of the evolution system (the case of two first integrals is also considered in this paper). According to variational methods, to show the stability of such solitary waves we have to show that the critical point is actually a local minimizer of the corresponding constrained variational problem. The major difficulty in applying this abstract method is to verify certain qualitative properties of the spectrum of the selfadjoint linearized operator M, which is the second derivative of the augmented Lagrangian at the critical point. In the examples we consider in this paper, the linearized operator M will be a linear selfadjoint ordinary differential operator given by a 2 × 2 system. Our main result states that, under some conditions, M has zero as a simple eigenvalue and it has exactly one negative eigenvalue. These are precisely the qualitative properties needed for the stability analysis using variational methods. We apply this result to study the stability of solitary waves for the following systems: a Schrodinger system from nonlinear optics (the so-called χ2-SHG equations), the (nonintegrable) Hirota–Satsuma system and a coupled system of a Schrodinger equation and a KdV equation.

A constrained minimization problem with integrals on the entire space

In this paper we consider the question of minimizing functionals defined by improper integrals. Our approach is alternative to the method of concentration-compactness and it does not require the verification of strict subaddivity.

On the structure of the spectrum of a linear time periodic wave equation

AbstractIn this paper we study the structure of the spectrum of the evolution operator generated by the time periodic wave equation

FitzHugh-Nagumo system: Boundedness and convergence to equilibrium

u_u = \frac{\partial }{{\partial x}}a(t,x)u_x ,

On the essential spectrum of a semigroup of thermoelasticity

with 2π periodicity inx as boundary condition. We show that such structure depends heavily on the rationality or irrationality of the rotation numbers of the characteristics deiffential equations