Differential and Integral Equations

Asymptotics for model nonlinear nonlocal equations

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin, and Isahi Sánchez-Suárez

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Differential Integral Equations, Volume 18, Number 11 (2005), 1273-1298.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35K15: Initial value problems for second-order parabolic equations


Hayashi, Nakao; Kaikina, Elena I.; Naumkin, Pavel I.; Sánchez-Suárez, Isahi. Asymptotics for model nonlinear nonlocal equations. Differential Integral Equations 18 (2005), no. 11, 1273--1298. https://projecteuclid.org/euclid.die/1356059742

Export citation