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, $A:D(A)subset Xto X$ is an unbounded sectorial operator with compact resolvent, and for some $lambda_0>0$ we have $mbox{Re}sigma(A)>lambda_0$ and $f:D(A^{frac13})subset Xto X$ is a nonlinear function with suitable conditions of growth and regularity.