Long-Time Existence for Semi-linear Beam Equations on Irrational Tori

We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n- 1 with n≥ 3 and d≥ 2. If ε≪ 1 is the size of the initial datum, we prove that the lifespan Tε of solutions is O(ε-A(n-2)-) where A≡A(d,n)=1+3d-1 when n is even and A=1+3d-1+max(4-dd-1,0) when n is odd. For instance for d= 2 and n= 3 (quadratic nonlinearity) we obtain Tε=O(ε-6-), much better than O(ε- 1) , the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of Δ2+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.