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 \(\varepsilon \ll 1\) is the size of the initial datum, we prove that the lifespan \(T_\varepsilon \) of solutions is \(O(\varepsilon ^{-A(n-2)^-})\) where \(A\equiv A(d,n)= 1+\frac{3}{d-1}\) when n is even and \(A= 1+\frac{3}{d-1}+\max (\frac{4-d}{d-1},0)\) when n is odd. For instance for \(d=2\) and \(n=3\) (quadratic nonlinearity) we obtain \(T_\varepsilon =O(\varepsilon ^{-6^-})\), much better than \(O(\varepsilon ^{-1})\), 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.

我们考虑d维无理环上的半线性束方程，其光滑非线性为\（n-1 \）与\（nge 3 \）和\（dge 2 \）。如果\（\ varepsilon \ ll 1 \）是初始基准的大小，我们证明溶液的寿命\（T_ \ varepsilon \）为\（O（\ varepsilon ^ {-A（n-2）^- }} \）其中\（A \ equiv A（d，n）= 1+ \ frac {3} {d-1} \）当n为偶数且\（A = 1+ \ frac {3} {d-当n为奇数时，1} + \ max（\ frac {4-d} {d-1}，0）\）。例如，对于\（d = 2 \）和\（n = 3 \）（二次非线性），我们得到\（T_ \ varepsilon = O（\ varepsilon ^ {-6 ^-}）\），比局部存在理论给出的时间（\（O（\ varepsilon ^ {-1}）\）好得多。圆环的非理性使得\（\ sqrt {\ Delta ^ 2 + 1} \）的两个特征值之间的差集累积为零，从而促进了高傅立叶模态之间的交换，并使长时间控制解变得复杂。我们的结果是通过组合Birkhoff法线形式步骤和修改后的能量步骤而获得的。