Asymptotics for model nonlinear nonlocal equations

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.