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

Abstract

Abstract 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 $${\mathscr {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> . Under suitable assumptions on the decay at infinity of the imaginary parts of the coefficients of P we prove the existence of a solution with the same Gevrey regularity of the data and we describe its behavior for $$|x| \rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> .