Long-time existence for semi-linear beam equations on irrational tori

TL;DR

This paper proves long-time existence of solutions for semi-linear beam equations on irrational tori, showing solutions persist much longer than local theory predicts, due to the torus's irrationality affecting eigenvalue differences.

Contribution

It introduces a novel analysis combining Birkhoff normal form and modified energy methods to establish extended lifespan results for semi-linear beam equations on irrational tori.

Findings

Lifespan Tε scales as ε^(-A(n-2)), with A depending on dimension and nonlinearity.

For d=2, n=3, lifespan improves to O(ε^(-6)), surpassing local existence bounds.

Irrationality of the torus influences eigenvalue differences, enabling longer solution lifespans.

Abstract

We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n -- 1 with n $\ge$ 3 and d $\ge$ 2. If $\epsilon$ $\ll$ 1 is the size of the initial datum, we prove that the lifespan T$\epsilon$ of solutions is O($\epsilon$ --A(n--2) --) where A $\not\equiv$ A(d, n) = 1 + 3 d--1 when n is even and A = 1 + 3 d--1 + max(4--d d--1 , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) we obtain T$\epsilon$ = O($\epsilon$ --6 --), much better than O($\epsilon$ --1), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of $\sqrt$ $\Delta$ 2 + 1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step.