The Cauchy problem for $p$-evolution equations with initial data in Gelfand-Shilov spaces

We study the Cauchy problem for a class of linear evolution equations of arbitrary order with coefficients depending both on time and space variables. Under suitable decay assumptions on the coefficients of the lower order terms for $\mathrm{\mid }x\mathrm{\mid }$ large, we prove a well-posedness result in Gelfand-Shilov spaces.