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

Abstract

In this paper, we discuss the well-posedness of the Cauchy problem associated with the third-order evolution equation in time $$u_{ttt}+Au+\eta A^{\frac{1}{3}}{u}_{tt}+\eta A^{\frac{2}{3}}{u}_t=f(u)$$ 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 $\mbox{Re}σ(A)>λ_0$ and $f:D(A^{\frac{1}{3}})\subset X\to X$ is a nonlinear function with suitable conditions of growth and regularity.

26 pages, 9 figures