Differential and Integral Equations

Asymptotics for model nonlinear nonlocal equations

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

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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

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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

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