Differential and Integral Equations

Subcritical pseudodifferential equation on a half-line with nonanalytic symbol

Elena I. Kaikina

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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


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