Related papers: Cauchy Problem for for some high order generalizat…
This article contains a new discussion for the generalized fractional Cauchy-type problem involving Hilfer-Katugampola-type fractional derivative. We study an existence and continuation of its solution. Firstly, we establish a new theorems…
This paper is devoted to studying the Cauchy problem for the Ostrovsky equation \begin{eqnarray*} \partial_{x}\left(u_{t}-\beta \partial_{x}^{3}u +\frac{1}{2}\partial_{x}(u^{2})\right) -\gamma u=0, \end{eqnarray*} with positive $\beta$ and…
The purpose is to study the Cauchy problem for non-linear in time and space pseudo-differential equations. These include the fractional in time versions of HJB equations governing the controlled scaled CTRW. As a preliminary step which is…
In the second part of this series of papers, we address the same Cauchy problem that was considered in part 1, namely the nonlocal Fisher-KPP equation in one spatial dimension, \[ u_t = D u_{xx} + u(1-\phi*u), \] where $\phi*u$ is a spatial…
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…
In this note, we improve a previously proven non-solvability result of the Cauchy problem for the Cauchy problem in the Gevrey class for a homogeneous second-order differential operator mentioned in the title. We prove that the Cauchy…
In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional…
We consider a linear non-autonomous evolutionary Cauchy problem \begin{equation} \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator $A(t)$ arises from a time depending…
We consider the Cauchy problem for the nonlinear Schr\"odinger equation on $\mathbb{R}^2$, $iu_t + u_{xx} + u_{yy} + \lambda|u|^\sigma u =0$, $\lambda\in \mathbb{R}$, $\sigma>0$. We introduce new functional spaces over which the initial…
This paper focuses on Cauchy problem for the three-dimensional two-fluid type model, in which the presence of vacuum is permitted. Under some assumptions that the initial data satisfy appropriate regularity conditions and a compatibility…
The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions…
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in the modulation space $M_{p,q}^{s}(\mathbb R)$ where $1\leq q\leq2$, $2\leq p<\frac{10q'}{q'+6}$ and…
In this paper we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta u=\lambda|x|^{-\alpha}|u|^{\beta}u$ in $H^1$. The well-posedness theory in $H^1$ has been intensively…
In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problem $$ \partial_t u+(-\Delta)^{\frac{\theta}{2}}u=u^p,\quad x\in{\bf R}^N,\,\,t>0, \qquad u(0)=\mu\ge…
In this paper, we state with a variational method a general theorem providing the existence of a weak solution $u$ for fractional Euler-Lagrange equations of the type: $$ \dfrac{\partial L}{\partial x} (u,D^\alpha_- u,t) + D^\alpha_+…
We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with vacuum as far-field density. It is proved that if the initial density decays not too slow at infinity, the 2D…
We investigate the Cauchy problem for a 2x2-system of weakly coupled semi-linear fractional wave equations with polynomial nonlinearities posed in R+ x RN. Under appropriate conditions on the exponents and the fractional orders of the time…
In this paper, we are dealing with the solution of the functional equation $$ \varphi\Big(\frac{x+y}2\Big)(f(x)-f(y))=F(x)-F(y), $$ concerning the unknown functions $\varphi,f$ and $F$ defined on a same open subinterval of the reals.…
In this paper, we discuss the existence and uniqueness of solutions of a boundary value problem for a fractional differential equation of order $\alpha\in(2,3)$, involving a general form of fractional derivative. First, we prove an…
In this article the unique solution of the Cauchy problem is founded by the Riemann method. Some relations for given here confluent hypergeometric functions of two and three variables are used.