## Differential and Integral Equations

- Differential Integral Equations
- Volume 18, Number 12 (2005), 1341-1370.

### Subcritical pseudodifferential equation on a half-line with nonanalytic symbol

#### Abstract

We study nonlinear pseudodifferential equations on a half-line with a nonanalytic symbol \begin{equation*} \left\{ \begin{array}{c} \partial _{t}u+\mathbb{K}u=\lambda \left\vert u\right\vert ^{\sigma }u,\text{ }x\in \mathbf{R}^{+},\text{ }t>0, \\ u\left( 0,x\right) =u_{0}\left( x\right) \text{, }x\in \mathbf{R}^{+}, \end{array} \right. \end{equation*} where $0<$ $\sigma <1,$ $\lambda \in \mathbf{R}$ and \begin{equation*} \mathbb{K}u=\frac{1}{2\pi i}\theta (x)\int_{-i\infty }^{i\infty }e^{px}K(p) \widehat{u}(t,p)dp,\qquad K(p)=\frac{p^{2}}{p^{2}-1}. \end{equation*} The aim of this paper is to prove the global existence of solutions to the initial-boundary-value problem and to find the main term of the asymptotic representation of solutions in subcritical case, when the nonlinear term of equation has the time decay rate less than that of the linear terms.

#### Article information

**Source**

Differential Integral Equations, Volume 18, Number 12 (2005), 1341-1370.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356059714

**Mathematical Reviews number (MathSciNet)**

MR2174976

**Zentralblatt MATH identifier**

1212.35524

**Subjects**

Primary: 35S15: Boundary value problems for pseudodifferential operators

Secondary: 35B40: Asymptotic behavior of solutions 35C20: Asymptotic expansions

#### Citation

Kaikina, Elena I. Subcritical pseudodifferential equation on a half-line with nonanalytic symbol. Differential Integral Equations 18 (2005), no. 12, 1341--1370. https://projecteuclid.org/euclid.die/1356059714