We study the Cauchy problem \begin{equation} \label{CP} \begin{cases} P(t,x,D_t,D_x)u(t,x) =f(t,x) \\ u(0,x)=g(x) \end{cases}, \qquad (t,x) \in [0,T] \times \mathbb{R}, \end{equation} for $p$-evolution operators of the form $$P(t,x,D_t,D_x)= D_t + a_p(t) D_x^p + \sum_{j=1}^{p-1} a_j(t,x)D_x^j, \qquad (t,x) \in [0,T]\times \mathbb{R},$$ where $a_p \in C([0,T], \mathbb{R})$ and $a_j \in C([0,T], C^\infty(\mathbb{R}; \mathbb{C})), j=0,\ldots,p-1,$ in the Gevrey functional setting. When the coefficients $a_j(t,x), j=0,\ldots,p-1,$ of the lower order terms are complex-valued, it is possible to obtain well-posedness results in Gevrey spaces under suitable decay assumptions on $a_j$ for $|x| \to \infty.$ In the first part of the talk, we present a well-posedness result for $3$-evolution equations obtained in [1]. In the second part we discuss necessary conditions for Gevrey well-posedness in the case of $p$-evolution equations for an arbitrary positive integer $p$, see [2]. The results presented in the talk are obtained in collaboration with Alexandre Arias Junior (Universit\`{a} di Torino) and Alessia Ascanelli (Universit\`{a} di Ferrara). References: [1] A. Arias Junior, A. Ascanelli, M. Cappiello, \textit{Gevrey well-posedness for $3$-evolution equatons with variable coefficients}, 2022. To appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci. DOI: 10.2422/2036-2145.202202\_011, https://arxiv.org/abs/2106.09511; [2]A. Arias Junior, A. Ascanelli, M. Cappiello, {\it On the Cauchy problem for $p$-evolution equations with variable coefficients: a necessary condition for Gevrey well-posedness}. Preprint (2023), https://arxiv.org/abs/2309.05571