相关论文: On the Cauchy problem for $D_t^2-D_x(b(t)a(x))D_x$

We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variables and random stationary ergodic in time. As was proved in [24] and [12] in this case…

We consider an evolution equation with the Caputo-Dzhrbashyan fractional derivative of order $\alpha \in (1,2)$ with respect to the time variable, and the second order uniformly elliptic operator with variable coefficients acting in spatial…

We study the wellposedness of Cauchy problem for the fourth order nonlinear Schr\"odinger equations i\partial_t u=-\eps\Delta u+\Delta^2 u+P((\partial_x^\alpha u)_{\abs{\alpha}\ls 2}, (\partial_x^\alpha \bar{u})_{\abs{\alpha}\ls 2}),\quad…

We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity in the periodic setting. Our main hypotheses are both that the dispersive operator behaves for high frequencies as a Fourier multiplier by $…

It is shown that for a function $f:\mathbb R^2\to \mathbb R$ which is measurable with respect to the first variable and upper semicontinuous quasicontinuous and increasing with respect to the second variable there exists a Caratheodory's…

We consider the Cauchy problem for the generalized fractional Korteweg-de Vries equation $$ u_t+D^\alpha u_x + u^p u_x= 0, \quad 1<\alpha\le 2, \quad p\in {\mathbb N}\setminus\{0\}, $$ with homogeneous initial data $\Phi$. We show that,…

For the Cauchy problem for an operator differential equation of the form $y'(z) = Ay(z)$, where $A$ is a closed linear operator on a Banach space over the completion of an algebraic closure of the field of $p$-adic numbers, a criterion of…

The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate…

We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variable and random stationary ergodic in time. As was proved in [25] and [13] in this case…

For an operator-differential equation of the form $y^{(m)}(z) = Ay(z)$, where $A$ is a closed linear operator on a Banach space over the the field of $p$-adic numbers, the necessary and sufficient conditions on initial data for the Cauchy…

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…

The abstract first order Cauchy problem is solved in terms of Taylor's series leading to a series of operators which is a propagator. It is found that higher order Cauchy problems can be solved in the same way. Since derivatives of order…

We study a class of higher-order KdV equations. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when in H^s(\R), for any real…

The Cauchy-problem for the generalized Kadomtsev-Petviashvili-II equation $$u_t + u_{xxx} + \partial_x^{-1}u_{yy}= (u^l)_x, \quad l \ge 3,$$ is shown to be locally well-posed in almost critical anisotropic Sobolev spaces. The proof combines…

To every Darboux integrable system there is an associated Lie group $G$ which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial…

Considered is the generalized Korteweg-de Vries-Burgers equation $$ u_{t}+u_{xxx}+uu_{x}+|D_{x}|^{2\alpha}u=0,\quad t\in \mathbb{R}^{+}, x\in \mathbb{R}, $$ with $0\leq \alpha\le 1$. We prove a sharp results on the associated Cauchy problem…

We study effectively hyperbolic operators $P$ with triple characteristics points lying on $t= 0$. Under some conditions on the principal symbol of $P$ one proves that the Cauchy problem for $P$ in $[0, T] \times U$ is well posed for every…

We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet…

We establish a rigorous link between infinite-dimensional regular Fr\"olicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili…

As is known, the problems for the differential equations with continuously changing order of the derivatives are not considered completely. In this paper we consider the initial and boundary value problems for this type of linear ordinary…