This paper addresses the problem of global well-posedness of a coupled system of Korteweg–de Vries equations, derived by Majda and Biello in the context of nonlinear resonant interaction of Rossby waves, in a periodic setting in homogeneous Sobolev spaces Ḣs, for s≥0. Our approach is based on a successive time-averaging method developed by Babin, Ilyin and Titi [A.V. Babin, A.A. Ilyin and E.S. ...

In this paper we study the local well-posed integrated Cauchy problem, v′(t) = Av(t) + tα Γ(α + 1) x, v(0) = 0, t ∈ [0, κ), with κ > 0, α ≥ 0, and x ∈ X where X is a Banach space and A a closed operator on X. We extend solutions increasing the regularity in α. The global case (κ = ∞) is also treated in detail. Growths of solutions are given in both cases.

We report on new results concerning the global well-posedness, dissipativity and attractors of the damped quintic wave equations in bounded domains of R.

This paper is devoted to the numerical analysis of ill-posed problems of evolution equations in Banach spaces using certain classes of stochastic one step methods. The linear stability properties of these methods are studied. Regularisation is given by the choice of the regularisation parameter as = p n ; where n is the stepsize and provides the convergence on smooth initial data. The case of t...

A novel multisensor approach to deconvolution is developed. This theory circumvents the ill-posedness inherent in convolution equations by overdetermining the input signal by a multichannel system of convolvers f ig, chosen so that any information lost by one channel is retained by another. The deconvolution problem is then solved by constructing \deconvolvers" that allow us to construct the Di...

We consider the system of dual-phase-lag thermoelasticity proposed by Chandrasekharaiah and Tzou. First, we prove that the solutions of the problem are generated by a semigroup of quasi-contractions. Thus, the problem of third-order in time is well-posed. Then the expontial stability is investigated. Finally the spatial behavior of solutions is analyzed in a semi-infinite cylinder and a resuult...

We prove that the generalized Benjamin-Ono equations ∂tu + H∂2 xu ± u ∂xu = 0, k ≥ 4 are locally well-posed in the scaling invariant spaces Ḣsk(R) where sk = 1/2 − 1/k. Our results also hold in the nonhomogeneous spaces Hsk(R). In the case k = 3, local well-posedness is obtained in Hs(R), s > 1/3.

In this paper, adaptive discrete-time low-gain integral control strategies are presented for tracking constant reference signals for infinite-dimensional discrete-time power-stable linear systems. The discrete-time results are applied in the development of adaptive sampled-data low-gain integral control of well-posed infinite-dimensional exponentially stable linear systems. Our results consider...

The paper studies a class of conservation law models for traffic flow on a family of roads, near a junction. A Riemann Solver is constructed, where the incoming and outgoing fluxes depend Hölder continuously on the traffic density and on the drivers’ turning preferences. However, various examples show that, if junction conditions are assigned in terms of Riemann Solvers, then the Cauchy problem...

We prove the global well-posedness and we study the linear response for a system of two coupled equations composed of a Dirac equation for an infinite rank operator and a nonlinear wave or Klein-Gordon equation.