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

Ya. I. Alber

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arXiv · funct-an/9312006

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

Ya. I. Alber

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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

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Ya. I. Alber. 1994-01-02. A New Approach to Investigation of Evolution Differential Equations in Banach Spaces. https://arxiv.org/abs/funct-an/9312006

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