Spectral Analysis for Some Third-Order Differential Equations: Well-Posedness and Regularity

Abstract

In this article, we present a spectral analysis for a class of abstract evolution equations of third order in time in order to study the well-posedness of these equations. Namely, we consider the following class of differential equations $$\begin{array}{r}{\displaystyle \frac{{\mathrm{d}}^3\mathrm{u}}{\mathrm{d}{t}^3}}+a{A}^{\frac{1}{3}}{\displaystyle \frac{{\mathrm{d}}^{2}u}{\mathrm{d}{t}^2}}+b{A}^{\frac{2}{3}}{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}t}}+cAu=0\end{array}$$ where $a,b,c$ are non-negative constants, $X$ is a Banach space, $-A:D(A)\subset X\to X$ is the infinitesimal generator of an exponentially stable semigroup in $X$ and $A^{\alpha }$ denotes the fractional power of $A$ . We determine, in terms of the coefficients $a,b,c$ , when this equation is either ill-posed, or hyperbolic, or parabolic, in the sense of the theory of strongly continuous semigroups of bounded linear operators. Additionally, we present an example using the fractional Laplacian on a bounded smooth domain of ${\mathbb{R}}^N$ .