Long time dynamics of the 𝑢^{𝑝} Klein-Gordon equation on general rectangular tori

We address two interesting questions in the theory of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Superscript p"> <mml:semantics> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">u^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Klein-Gordon equation defined on the general rectangular tori. First, we show the existence and uniqueness of solutions using an explicit combinatorial analysis under the exponential decay assumption in the Fourier space. The second question is the long time behavior of such a solution. In a weakly nonlinear setting, we can prove that the nonlinear solution is asymptotic to the linear solution.