We obtain various new well-posedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of vector fields with bounded compression and possibly having a blow-up of the BV norm at the initial time. We apply these results, valid in any space dimension, to a 2×2 system of conservation laws in one...

In this paper we prove that the initial value problem associated to the following higher-order Benjamin-Ono equation ∂tv − bH∂ xv + a∂ xv = cv∂xv − d∂x(vH∂xv + H(v∂xv)), where x, t ∈ R, v is a real-valued function, H is the Hilbert transform, a ∈ R, b, c and d are positive constants, is locally well-posed for initial data v(0) = v0 ∈ H(R), s ≥ 2 or v0 ∈ H(R) ∩ L(R; xdx), k ∈ Z+, k ≥ 2.

The authors havc bccn st.udying color cstimation of gray-scale imagc. The prohlcm of this study is defined as an ill-posed problem. bccausc a luminancc to arhitrary color is uniquely fixed but the Illminancr c.orrcsponds t.o plural color-values. In this papcr. tintingcomponcnt (m whit.c mixed valuc) cst,imat ion mct hod from gray-scale imcgc is proposcd based on st.atist ics. This mcthod acquir...

The aim of the present paper is to obtain common fixed point theorems by employing the recently introduced notion of conditional reciprocal continuity. We demonstrate that conditional reciprocal continuity ensures the existence of fixed points under contractive conditions which otherwise do not ensure the existence of fixed points. Our results generalize and extend several well-known fixed poin...

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...