A New Approach to Investigation of Evolution Differential Equations in Banach Spaces

TL;DR

This paper investigates nonlinear evolution differential equations in Banach spaces without the usual Hilbert space embedding assumptions, establishing new stabilization results for solutions in this more general setting.

Contribution

It introduces a novel approach to analyze the stabilization of solutions of evolution equations in Banach spaces lacking Hilbert space structure.

Findings

Established stabilization of solutions without Hilbert space embedding.

Proved convergence in both weak and strong senses under new conditions.

Extended the theory to more general Banach space settings.

Abstract

Known investigations of nonlinear evolution equations $${dx\over dt} + A(t)x(t) = f(t)\ ,\quad x(t_{0}) = x^{0},\ \quad t_{0} \le t < \infty\ , \eqno(0.1)$$ with monotone operators $A(t)$ acting from reflexive Banach space $B$ to dual space $B^*$, usually assume that along with $B$ and $B^*$ there is a Hilbert space $H$ and continuous imbedding $B \hookrightarrow H$ in the triplet $$B \hookrightarrow H \hookrightarrow B^*\ ; \eqno(0.2)$$ and that $B$ is dense in $H$. The stabilization of solutions of evolution equations has been proven either in the sense of weak convergence in $B$ or in the norm of $H$ space, and only asymptotic estimates of stabilization rate have been obtained [15]. In the present paper we consider equations of type (0.1) without conditions (0.2) and establish stabilization with both