Qualitative analysis of evolution equations: Weakly continuous semigroups in Banach spaces

Abstract

Evolution equations and operator semigroups in Banach spaces play a pivotal role across various branches of applied mathematics. This paper focuses on the qualitative analysis of evolution equations, particularly first-order linear partial differential equations (PDEs) with Cauchy data and hyperbolic initial value problems, using weakly continuous semigroups. Leveraging the theory of weakly continuous semigroups of contractions, we establish fundamental theorems such as the Lumer-Phillips and Hill-Yosida theorems, which provide crucial insights into the generation of semigroups in Banach spaces. Additionally, we analyze the qualitative properties of solutions, addressing aspects of existence, uniqueness, and stability. Our findings deepen the understanding of solution behaviors in these specific contexts, bridging the theoretical framework of operator semigroup theory with practical applications in the study of evolution equations.