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On the Long Time Behavior of Solutions to the Intermediate Long Wave Equation

Research paper by Claudio Mun͂oz, Gustavo Ponce, Jean-Claude Saut

Indexed on: 13 Apr '21Published on: 16 Feb '21Published in: SIAM Journal on Mathematical Analysis



Abstract

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 $\beta\in{\mathbb{R}}$.