Well-posedness for some third-order evolution differential equations: A semigroup approach

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.