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

In this paper, we discuss the well-posedness of Cauchy problem associated with third-order evolution equation in time $$\begin{aligned} u_{ttt} +A u + \eta A^{\frac{1}{3}} u_{tt} +\eta A^{\frac{2}{3}} u_t=f(u) \end{aligned}$$ where $$\eta >0$$ , X is a separable Hilbert space, $$A:D(A)\subset X\rightarrow X$$ an unbounded sectorial operator compact resolvent, and for some $$\lambda _0>0$$ have $$\text{ Re }\sigma (A)>\lambda _0$$ $$f:D(A^{\frac{1}{3}})\subset nonlinear function suitable conditions growth regularity.