The Cauchy problem for 3-evolution equations with data in Gelfand-Shilov spaces

We consider the Cauchy problem for a third-order evolution operator P with (t, x)-depending coefficients and complex-valued lower-order terms. We assume the initial data to be Gevrey regular with an exponential decay at infinity, that is, the data belong to some Gelfand-Shilov space of type S. Under suitable assumptions on the decay at infinity of the imaginary parts of the coefficients of P we prove the existence of a solutionwith the sameGevrey regularity of the data andwe describe its behavior for vertical bar x vertical bar ->infinity.