Related papers: Local and global analyticity for $\mu$-Camassa-Hol…

The purpose of this paper is to investigate the Cauchy problem for the Gross-Pitaevskii infinite linear hierarchy of equations on $\mathbb{R}^n,$ $n \geq 1.$ We prove local existence and uniqueness of solutions in certain Sobolev type…

We study existence, uniqueness, and distributional aspects of generalized solutions to the Cauchy problem for first-order symmetric (or Hermitian) hyperbolic systems of partial differential equations with Colombeau generalized functions as…

We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.

We consider the Cauchy problem of massless Dirac-Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be…

In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold $M$: \begin{equation*} \begin{cases} u_{t}-\mathfrak{L}_{M} u=f(u), \;x\in M, \;t>0, \\u(0,x)=u_{0}(x), \;x\in M, \end{cases}…

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…

In the present paper, we study the Cauchy-Dirichlet problem to the nonlocal nonlinear diffusion equation with polynomial nonlinearities $$\mathcal{D}_{0|t}^{\alpha…

This work is devoted to the study of a nonlocal-in-time evolutional problem for the first order differential equation in Banach space. Our primary approach, although stems from the convenient technique based on the reduction of a nonlocal…

We study the Cauchy problem for the fractional semilinear heat equation with distributional inhomogeneous terms. By introducing the Lorentz--Morrey spaces, we overcome limitations of real interpolation in the classical local Morrey spaces…

We study the Cauchy problem for a system of semi-linear coupled fractional-diffusion equations with polynomial nonlinearities posed in $% \mathbb{R}_{+}\times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the orders of…

Brief account of results on the Cauchy problem for the Einstein equations starting with early the works of Darmois and Lichnerowicz and going up to the proofs of the existence and uniqueness of solutions global in space and local in time,…

In this paper, we consider the Cauchy problem of the multi-dimensional generalized MHD system in the whole space and construct global smooth solutions with a class of large initial data by exploring the structure of the nonlinear term.…

In this paper, the global smooth solution of Cauchy's problem of incompressible, resistive, viscous Hall-magnetohydrodynamics (Hall-MHD) is studied. By exploring the nonlinear structure of Hall-MHD equations, a class of large initial data…

We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with step-like initial data \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}=0, \quad m=u-u_{xx}, \nonumber \\…

We consider the Cauchy problem for strictly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is $L^2$ well-posed in the case of Lipschitz…

We study the Cauchy problem for the quasi-geostrophic equations with the critical dissipation in the two dimensional half space under the homogeneous Dirichlet boundary condition. We show the global existence, the uniqueness and the…

We study the Cauchy problem of higher dimensional Einstein-Maxwell-Higgs system in the framework of Bondi coordinates. As a first step, the problem is reduced to a single first-order integro-differential equation by defining a generalized…

The integrable Novikov equation can be regarded as one of the Camassa-Holm-type equations with cubic nonlinearity. In this paper, we prove the global existence and uniqueness of the H\"older continuous energy conservative solutions for the…

We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in $L^2$. Moreover, we prove that the solution exists…

The paper investigates the initial problem for a linear system of ordinary differential equations with the fractional differentiation operator Dzhrbashyan -- Nersesyan with constant coefficients. The existence and uniqueness theorems of the…