The Very Fast Solution of a Special Second Order ODE with Exponentially Decaying Forcing and Applications

Haraux, Alain

Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 2, pp. 233-241

Résumé

Let $b$, $c$, $p$ be arbitrary positive constants and let $f \in C(\mathbb{R}^{+})$ be such that for some $\lambda > c$, $F > 0$ we have $|f(t)| \leq F \exp(-\lambda t)$. Then all solutions $x$ of \begin{equation*} \tag{E} x'' + cx' + b|x|^{p}x = f(t) \end{equation*} tend to 0 as well as $x'$ as $t$ tends to infinity. Moreover there exists a unique solution $y$ of (E) such that for some constant $C > 0$ we have $|y(t)| + |y'(t)| \leq C \exp(-\lambda t)$ for all $t > 0$. Finally all other solutions of (E) decay to 0 either like $e^{-ct}$ or like $(1+t)^{-1/p}$ as $t$ tends to infinity.