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

TL;DR

This paper establishes the well-posedness of a third-order evolution differential equation in a Hilbert space using a semigroup approach, considering nonlinearities and specific operator conditions.

Contribution

It provides new results on the existence, uniqueness, and continuous dependence of solutions for a class of third-order evolution equations with nonlinear terms.

Findings

Proved well-posedness under certain operator and nonlinearity conditions

Extended semigroup methods to third-order evolution equations

Analyzed the impact of the operator's spectrum on solution behavior

Abstract

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.