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

Abstract We consider the Cauchy problem for a third-order evolution operator P with ( t , x )-depending coefficients and complex-valued lower-order terms. assume initial data to be Gevrey regular an exponential decay at infinity, that is, belong some Gelfand–Shilov space of type $${\mathscr {S}}$$ S . Under suitable assumptions on infinity imaginary parts we prove existence solution same regularity describe its behavior $$|x| \rightarrow \infty $$ | x ? ?