Complicated Asymptotic Behavior of Solutions for Heat Equation in Some Weighted Space

Abstract

We investigate the asymptotic behavior of solutions for the heat equation in the weighted space $Y_0^{\sigma }({ℝ}^N)\equiv \{\phi \in C({ℝ}^N):{\lim \hspace{0.17em}}_{|x|\to \infty }(1+|x{|}^2{)}^{-\sigma /2}\phi (x)=0\}$. Exactly, we find that the unbounded function space $Y_0^{\sigma }({ℝ}^N)$ with can provide a setting where complexity occurs in the asymptotic behavior of solutions for the heat equation.