Prescribing Oscillation Behavior of Solutions to the Heat Equation on $\mathbb{R}^n$ via the Initial Data and its Average Integral

Abstract

\begin{abstract} Motivated by a classical stabilization result for solution to the Cauchy problem of the heat equation$\ \partial_{t}u=\bigtriangleup u\ $on $\mathbb{R}^{n}$, we consider its oscillation behavior with radial initial data $φ\left( x\right) =φ\left( \left\vert x\right\vert \right) \in C^{0}\left( \mathbb{R}^{n}\right) \bigcap L^{\infty}\left( \mathbb{R}^{n}\right) .\ $Given four arbitrary finite numbers $r<α <β<s,$ one can construct a radial $φ\in C^{0}\left( \mathbb{R}% ^{n}\right) \bigcap L^{\infty}\left( \mathbb{R}^{n}\right) $ so that $φ $together with its corresponding solution$\ u\left( x,t\right) $ satisfy the oscillation behavior: \begin{align*} \liminf_{τ\rightarrow\infty}φ\left( τ\right) & =r<\liminf _{t\rightarrow\infty}u\left( 0,t\right) =α & <\limsup_{t\rightarrow\infty}u\left( 0,t\right) =β<\limsup _{τ\rightarrow\infty}φ\left( τ\right) =s. \end{align*} Another related topic concerning the oscillation behavior of the average integral of the initial data is also discussed. \end{abstract}