Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [3]. The same arguments may apply to obtain local wellposedness for the NL...

The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter.

The Cauchy problem for the Fokker–Plank–Kolmogorov equation with a nonlocal nonlinear drift term is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Fokker–Plank– Kolmogorov equations is considered. Illustrative examples of the one-dimensional symmetry operators are presented.

We consider a nonsymmetric algebraic matrix Riccati equation arising from transport theory. The nonnegative solutions of the equation can be explicitly constructed via the inversion formula of a Cauchy matrix. An error analysis and numerical results are given. We also show a comparison theorem of the nonnegative solutions.

We solve the inverse scattering problem for multidimensional vector fields and we use this result to construct the formal solution of the Cauchy problem for the second heavenly equation of Plebanski, a scalar nonlinear partial differential equation in four dimensions relevant in General Relativity, which arises from the commutation of multidimensional Hamiltonian vector fields.

In this paper we focus on the global-in-time existence and the pointwise estimates of solutions to the Cauchy problem for the semilinear dissipative wave equation in multi-dimensions. By using the method of Green function combined with the energy estimates, we obtain the pointwise estimates of the solution. keywords: semilinear dissipative wave equation, pointwise estimates, Green function. MSC...

We study the Cauchy initial-value problem for the Benjamin-Ono equation in the zero-dispersion limit, and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by Lax and Levermore to analyze the corresponding limit for the Korteweg–de Vries equation. © 2010 Wiley Periodicals, Inc.

The main results of this paper are concerned with the “bad” behavior of the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces. This leads to some kind of ill-posedness of the corresponding Cauchy problem: the flow map cannot be of class C2 in any Sobolev space.