The Cauchy problem for a first order system of abstract operator equations

Inhomogeneous Two-Parameter Abstract Cauchy Problem

Singularly Perturbed Cauchy Problem for Abstract Linear Differential Equations of Second Order in Hilbert Spaces∗

We study the behavior of solutions to the problem { ε (u′′ ε (t) +A1uε(t)) + u ′ ε(t) +A0uε(t) = fε(t), t ∈ (0, T ), uε(0) = u0ε, uε(0) = u1ε, as ε → 0, where A1 and A0 are two linear self-adjoint operators in a Hilbert space H. MSC: 35B25, 35K15, 35L15, 34G10 keywords: singular perturbations; Cauchy problem; boundary layer function.

On the Cauchy problem for higher-order nonlinear dispersive equations

We study the higher-order nonlinear dispersive equation ∂tu+ ∂ 2j+1 x u = ∑ 0≤j1+j2≤2j aj1,j2∂ j1 x u∂ j2 x u, x, t ∈ R. where u is a real(or complex-) valued function. 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 a0,k 6= 0 for some k > j, in the sense that this equation cann...

A singular abstract cauchy problem.

In this note a singular abstract Cauchy problem is considered. A solution is obtained for this problem in terms of a regular abstract Cauchy problem. As an application we obtain a new solution of the initial value problem for a class of singular partial differential equations.

Existence and blow-up of solution of Cauchy problem for the sixth order damped Boussinesq equation

‎In this paper‎, ‎we consider the existence and uniqueness of the global solution for the sixth-order damped Boussinesq equation‎. ‎Moreover‎, ‎the finite-time blow-up of the solution for the equation is investigated by the concavity method‎.