We show that for any uniformly bounded in time \(H^1\cap L^1\) solution of the dispersion generalized Benjamin–Ono equation, the limit infimum, as time t goes to infinity, converges to zero locally in an increasing-in-time region of space of order \(t/\log t\). This result is in accordance with the one established by Muñoz and Ponce (Proc Am Math Soc 147(12):5303–5312, 2019) for solutions of the Benjamin–Ono equation. Similar to solutions of the Benjamin–Ono equation, for a solution of the dispersion generalized Benjamin–Ono equation, with a mild \(L^1\)-norm growth in time, its limit infimum must converge to zero, as time goes to infinity, locally in an increasing on time region of space of order depending on the rate of growth of its \(L^1\)-norm. As a consequence, the existence of breathers or any other solution for the dispersion generalized Benjamin–Ono equation moving with a speed “slower” than a soliton is discarded. In our analysis the use of commutators expansions is essential.

我们证明，对于色散广义本杰明-奥诺方程的时间均匀约束的时间（（H ^ 1 \ cap L ^ 1 \）），随着时间t趋于无穷大，极限最小会以递增的方式局部收敛于零。\（t / \ log t \）的空间的及时区域。此结果与Muñoz和Ponce建立的结果（Proc Am Math Soc 147（12）：5303–5312，2019）建立的结果一致，用于解决Benjamin-Ono方程。与Benjamin–Ono方程的解相似，对于色散广义Benjamin–Ono方程的解具有温和\（L ^ 1 \）-范数随时间增长，它的极限最小值必须随着时间趋于无穷而收敛于零，这取决于其（L ^ 1 \）-范数的增长率。结果是，存在色散广义本杰明-奥诺方程以比孤子“慢”的速度运动的呼吸器或任何其他解决方案的存在。在我们的分析中，必须使用换向器扩展。