相关论文: Non-uniform dependence for a generalized Degasperi…

We prove that the Cauchy problem for the KP-I equation is globally well-posed for initial data which are localized perturbations (of arbitrary size) of a non-localized (i.e. not decaying in all directions) traveling wave solution (e.g. the…

We study the Cauchy problem for a general inhomogeneous linear moment partial differential equation of two complex variables with constant coefficients, where the inhomogeneity is given by the formal power series. We state sufficient…

We define and study the Cauchy problem for a 1-D nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including…

We study the well-posedness of the Cauchy problem for scalar conservation laws with discontinuous, non-degenerate fluxes. Locally, the fluxes are piecewise smooth across interfaces described by a Heaviside-type discontinuity, with left and…

The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as a model delineating the propagation of shallow water waves, stands as a completely integrable system…

This paper is devoted to the analysis of the incompressible Euler equation in a time-dependent fluid domain, whose interface evolution is governed by the law of linear elasticity. Our main result asserts that the Cauchy problem is globally…

We show the lack of uniform continuity of the flow map for the Camassa-Holm equation on the line, in the Sobolev spaces of index s > 3/2.

We consider the Cauchy problem for the generalized Kadomtsev-Petviashvili equations with the dissipation term $-\nu u_{xx}$ in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropy. In this…

Suppose that an $n$-dimensional Cauchy problem \frac{dx}{dt}=f(t,x,\mu) (t \in I, \mu \in M), x(t_0)=x^0 satisfies the conditions that guarantee existence, uniqueness and continuous dependence of solution x(t,t_0,\mu) on parameter \mu in an…

This paper is concerned with the Cauchy problem of a multivalued ordinary differential equation governed by the hypergraph Laplacian, which describes the diffusion of ``heat'' or ``particles'' on the vertices of hypergraph. We consider the…

We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending…

The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as an asymptotic approximation for the unidirectional propagation of shallow water waves, is an integrable…

This article represents a first step towards understanding the well-posedness for the dispersive Hunter-Saxton equation. This problem arises in the study of nematic liquid crystals, and although the equation has formal similarities with the…

The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…

These notes are devoted to the notion of well-posedness of the Cauchy problem for nonlinear dispersive equations. We present recent methods for proving ill-posedness type results for dispersive PDE's. The common feature in the analysis is…

In this paper, we study the solvability of a Cauchy- Dirichlet problem for nonlinear parabolic equation with non standard growths and nonlocal terms. We show the existence of weak solutions of the considered problem under more general…

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we…

This paper studies the properties of solutions for a double nonlinear time-dependent parabolic equation with variable density, not in divergence form with a source or absorption. The problem is formulated as a partial differential equation…

First, using the uniform decomposition in both physical and frequency spaces, we obtain an equivalent norm on modulation spaces. Secondly, we consider the Cauchy problem for the dissipative evolutionary pseudo-differential equation…

We consider the Cauchy problem for the equations of pressureless gases in two space dimensions. For a generic set of smooth initial data (density and velocity), it is known that the solution loses regularity at a finite time $t_0$, where…