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

Known investigations of nonlinear evolution equations $$\frac{dx}{dt}+A(t)x(t)=f(t),x(t_0)=x^0,t_0\le t<\mathrm{\infty },\text{eqno}(0.1)$$ with monotone operators $A(t)$ acting from reflexive Banach space $B$ to dual space $B^{\ast }$, usually assume that along with $B$ and $B^{\ast }$ there is a Hilbert space $H$ and continuous imbedding $B\hookrightarrow H$ in the triplet $$B\hookrightarrow H\hookrightarrow B^{\ast };\text{eqno}(0.2)$$ 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