Related papers: The Cauchy problem for $3$-evolution equations wit…
We address the problem of analyticity up to the boundary of solutions to the Euler equations in the half space. We characterize the rate of decay of the real-analyticity radius of the solution $u(t)$ in terms of $\exp{\int_{0}^{t} \Vert…
This work is devoted to the study of a nonlocal-in-time evolutional problem for the first order differential equation in Banach space. Our primary approach, although stems from the convenient technique based on the reduction of a nonlocal…
We study the following Cauchy problem for the linear wave equation with both time-dependent friction and time-dependent viscoelastic damping: \begin{equation} \label{EqAbstract}\tag{$\ast$} \begin{cases} u_{tt}- \Delta u + b(t)u_t -…
We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the…
In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface $\Sigma \simeq \overline{B(0,1)} \subset \mathbb{R}^3$ and…
In this paper, we deal with a Cauchy problem for a nonlinear fractional differential equation with the Caputo derivative of order $\alpha \in (0, 1)$. As initial data, we consider a pair consisting of an initial point, which does not…
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…
We investigate the Cauchy problem for elliptic operators with $C^\infty$-coefficients at a regular set $\Omega \subset R^2$, which is a classical example of an ill-posed problem. The Cauchy data are given at the subset $\Gamma \subset…
In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi linear $sigma$ evolution equations with different damping mechanisms for any $\sigma>1$, parabolic like damping and $\sigma$ evolution like…
We investigate the Cauchy problem for a heat equation driven by the mixed local-nonlocal operator $\mathcal{L}:=-\Delta+(-\Delta)^s$, $s\in(0,1)$, with exponential nonlinearity \[ \partial_tu(x,t)+\mathcal{L}u(x,t)=f(u(x,t)), \qquad…
In this paper we consider some dissipative versions of the modified Korteweg de Vries equation $u_t+u_{xxx}+|D_x|^{\alpha}u+u^2u_x=0$ with $0<\alpha\leq 3$. We prove some well-posedness results on the associated Cauchy problem in the…
The well-posedness of the abstract \textsc{Cauchy} problem for the doubly nonlinear evolution inclusion equation of second order \begin{align*} \begin{cases} u''(t)+\partial \Psi(u'(t))+B(t,u(t))\ni f(t), &\quad t\in (0,T),\, T>0,\\…
We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial…
The study is made of the problem of multiple interpolation on an infinite nodes set by the sums of absolutely convergent series of exponentials whose exponents are from a given set. For entire function conditions on nodes and exponents are…
It is shown that, if all weak solutions of the evolution equation \begin{equation*} y'(t)=Ay(t),\ t\ge0, \end{equation*} with a scalar type spectral operator $A$ in a complex Banach space are Gevrey ultradifferentiable of orders less than…
The resolution of a very large class of linear and non-linear, stationary and evolutive partial differential problems in the half-space (or similar) under the slip boundary condition is reduced here to that of the corresponding results for…
In this paper, we investigate the Cauchy problem associated to a system of PDE's of Oldroyd type. The considered model describes the evolution of certain viscoelastic fluids within a corotational framework. The non-corotational setting is…
We study the global well-posedness and asymptotic behavior of solutions for the Cauchy problem of three-dimensional sixth order Cahn-Hilliard equation arising in oil-water-surfactant mixtures. First, by using the pure energy method and a…
This Note derives regularity bounds for a Gevrey criterion when the Cauchy problem of elliptic equations is solved by regularization. When utilizing the regularization, one knows that checking such criterion is basically problematic, albeit…
An outstanding issue in the treatment of boundaries in general relativity is the lack of a local geometric interpretation of the necessary boundary data. For the Cauchy problem, the initial data is supplied by the 3-metric and extrinsic…