Related papers: A generalisation of the Cauchy-Kovalevskaia theore…

There is little analytical theory for the behavior of solutions of the Kuramoto-Sivashinsky equation in two spatial dimensions over long times. We study the case in which the spatial domain is a two-dimensional torus. In this case, the…

For perturbations of integrable Hamiltonians systems, the Nekhoroshev theorem shows that all solutions are stable for an exponentially long interval of time, provided the integrable part satisfies a steepness condition and the system is…

In this paper, for evolutionary Faddeev model corresponding to maps from the Minkowski space $\mathbb{R}^{1+n}$ to the unit sphere $\mathbb{S}^2$, we show the global nonlinear stability of geodesic solutions, which are a kind of nontrivial…

Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly…

This is a first version of a paper concerning abstract evolution equation with fractional time derivatives. Maximal regularity results in spaces of continuous and Hoelder continuous functions are described.

Using our results in [15], we provided existence theorems for the general classes of nonlinear evolutions. Finally, we give examples of applications of our results to parabolic, hyperbolic, Shr\"{o}dinger, Navier-Stokes and other…

A variant of the abstract Cauchy-Kovalevskaya theorem is considered. We prove existence and uniqueness of classical solutions to the nonlinear, non-autonomous initial value problem \[ \frac{du(t)}{dt} = A(t)u(t) + B(u(t),t), \ \ u(0) = x \]…

The singularity structure of solutions of a class of Hamiltonian systems of ordinary differential equations in two dependent variables is studied. It is shown that for any solution, all movable singularities, obtained by analytic…

Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of L_{p} spaces of functions whose regularity is defined by a Levy measure with O-regulary varying radial profile. Existence and uniqueness of a solution…

In this note we investigate the partial Fourier series on a product of two compact Lie groups. We give necessary and sufficient conditions for a sequence of partial Fourier coefficients to define a smooth function or a distribution. As…

We consider nonlinear equations having generalized Orlicz growth (also known as Musielak--Orlicz growth). We prove that if differential operators $\mathcal{A}_i$ converge locally uniformly to an operator $\mathcal{A}$, then the sequence of…

We describe a functional framework suitable to the analysis of the Cahn-Hilliard equation on an evolving surface whose evolution is assumed to be given \textit{a priori}. The model is derived from balance laws for an order parameter with an…

The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve…

Let $(-\Delta)_c^s$ be the realization of the fractional Laplace operator on the space of continuous functions $C_0(\mathbb{R})$, and let $(-\Delta_h)^s$ denote the discrete fractional Laplacian on $C_0(\mathbb{Z}_h)$, where $0<s<1$ and…

We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…

The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and…

In this paper, we study the Cauchy problem for an integrable multi-component (2N-component) peakon system which is involved in an arbitrary polynomial function. Based on a generalized Ovsyannikov type theorem, we first prove the existence…

We show some classes of higher order partial difference equations admitting a zero-curvature representation and generalizing lattice potential KdV equation. We construct integrable hierarchies which, as we suppose, yield generalized…

In this note we will discuss a potentially interesting extension of some recent results on primitive solutions to completely integrable partial differential equations. We will discuss a family distributions that are holomorphic on the…

For a class of evolution equations that possibly have only local solutions, we introduce a stochastic component that ensures that the solutions of the corresponding stochastically perturbed equations are global. The class of partial…