On Characterization of Solutions of Some Nonlinear Differential Equations and Applications
Marie‐Françoise Bidaut‐Véron, Mustapha Bouhar · SIAM Journal on Mathematical Analysis · 1994
This paper gives a complete classification of the solutions of the equation $\omega '' - \mu \omega + | \omega |^{q - 1} \omega = 0$ on $S^1 = {\mathbb{R} /{2\pi \mathbb{Z}}}$, where $\mu ,q \in \mathbb{R}$ with $q > 1$, and compares their different levels of energy. Thus the asymptotical behavior and all the possible orbits are found for the parabolic one-dimensional equation $u_t = u_{xx} + | u |^{q - 1} u - \mu u$ and the elliptic two-dimensional equation $\Delta u - c| x |^{ - 2} u + | u |^{q - 1} u = 0$, where $c > 0$.