SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 1029-1048, January 2021.
We show that the limit infimum, as time $\,t\,$ goes to infinity, of any uniformly bounded in time $H^{3/2+}\cap L^1$ solution to the intermediate long wave (ILW) equation converges to zero locally in an increasing-in-time region of space of order $\,t/\log(t)$. Also, for solutions with a mild $L^1$-norm growth in time it is established that its lim inf converges to zero, as time goes to infinity. This confirms the nonexistence of breathers and other solutions for the ILW model moving with a speed łqłq slower" than a soliton. We also prove that in the far field linearly dominated region, the $L^2$-norm of the solution also converges to zero as time approaches infinity. In addition, we deduced several scenarios for which the initial value problem associated to the generalized Benjamin--Ono and the generalized ILW equations cannot possess time periodic solutions (breathers). Finally, as previously demonstrated in solutions of the KdV and Benjamin--Ono equations, we establish the following propagation of regularity result: if the datum $u_0\in H^{3/2+}({\mathbb{R}})\cap H^m((x_0,\infty))$, for some $x_0\in{\mathbb{R}},\,m\in Z^+,\,m\geq 2$, then the corresponding solution $u(\cdot,t)$ of the ILW equation belongs to $H^m(\beta,\infty)$ for any $t>0$ and $\beta\in{\mathbb{R}}$.

中文翻译：

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关于中间长波方程解的长时间行为

SIAM数学分析杂志，第53卷，第1期，第1029-1048页，2021年1月。
我们显示，随着时间$ \，t \，$趋近于无穷大，任何在时间上$ H ^ {3/2 +} \ cap L ^ 1 $到中间长波（ILW）的统一有界的极限无穷大在阶次为\\，t / \ log（t）$的空间的递增区域中，方程局部收敛为零。同样，对于在时间上具有$ L ^ 1 $-范数温和增长的解决方案，可以确定的是，随着时间趋于无穷大，其lim inf收敛至零。这证实了ILW模型以比孤立子慢łqłq的速度运动的呼吸器和其他解决方案不存在。我们还证明，在远场线性控制区域中，该解决方案的$ L ^ 2 $-范数也收敛于当时间接近无穷大时为零。此外，我们推导了几种情形，其中与广义Benjamin-Ono和广义ILW方程相关的初始值问题无法拥有时间周期解（呼吸）。最后，如先前在KdV和Benjamin-Ono方程的解中所证明的，我们建立了以下规律性结果的传播：如果基准面$ u_0 \ in H ^ {3/2 +}（{\ mathbb {R}}） \ cap H ^ m（（x_0，\ infty））$，对于某些$ x_0 \ in {\ mathbb {R}}，\，m \ in Z ^ +，\，m \ geq 2 $，则对应的解决方案对于任何$ t> 0 $和$ \ beta \ in {\ mathbb {R}} $，ILW方程的$ u（\ cdot，t）$都属于$ H ^ m（\ beta，\ infty）$。