  In this paper, we discuss the well-posedness of the Cauchy problem associated
with the third-order evolution equation in time $$ u_{ttt} +A u + \eta
A^{\frac13} u_{tt} +\eta A^{\frac23} u_t=f(u) $$ where $\eta>0$, $X$ is a
separable Hilbert space, $A:D(A)\subset X\to X$ is an unbounded sectorial
operator with compact resolvent, and for some $\lambda_0>0$ we have
$\mbox{Re}\sigma(A)>\lambda_0$ and $f:D(A^{\frac13})\subset X\to X$ is a
nonlinear function with suitable conditions of growth and regularity.
