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2005 Asymptotics for model nonlinear nonlocal equations
Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin, Isahi Sánchez-Suárez
Differential Integral Equations 18(11): 1273-1298 (2005).

Abstract

We study the Cauchy problem for the model nonlinear equation \begin{equation} \left\{ \begin{array}{c} u_{t}+\mathcal{L}u=\lambda \left| u\right| ^{\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. \tag*{(0.1)} \end{equation} where $\sigma >0,$ $\lambda \in \mathbf{R.}$ We are interested in the critical and subcritical powers of the nonlinearity, especially in the case of large initial data $u_{0}$ from $\mathbf{L}^{1,a}\cap \mathbf{L}^{\infty }.$ We prove that the Cauchy problem (0.1) has a unique global solution $ u\in \mathbf{C}\left( [0,\infty );\mathbf{L}^{\infty }\cap \mathbf{L} ^{1,a}\right) $ and obtain the large time asymptotics.

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Nakao Hayashi. Elena I. Kaikina. Pavel I. Naumkin. Isahi Sánchez-Suárez. "Asymptotics for model nonlinear nonlocal equations." Differential Integral Equations 18 (11) 1273 - 1298, 2005.

Information

Published: 2005
First available in Project Euclid: 21 December 2012

zbMATH: 1212.35211
MathSciNet: MR2174821

Subjects:
Primary: 35K55
Secondary: 35B40 , 35K15

Rights: Copyright © 2005 Khayyam Publishing, Inc.

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Vol.18 • No. 11 • 2005
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