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The typical problem is: $$\\dot{u}=A(t)u+F(t,u), \\quad t\\geq 0; \\,\\, u(0)=u_0;\\quad \\dot{u}=\\frac {du}{dt}\\qquad (*)$$ Here $A(t)$ is a linear bounded operator in a Hilbert space $H$, and $F$ is a nonlinear operator, $\\|F(t,u)\\|\\leq c_0\\|u\\|^p,\\,\\,p>1$, $c_0, p=const>0$. It is assumed that Re$(A(t)u,u)\\leq -\\gamma(t)\\|u\\|^2$ $\\forall u\\in H$, where $\\gamma(t)>0$, and the case when $\\lim_{t\\to \\infty}\\gamma(t)=0$ is also considered. An estimate of the rate of decay of solutions to problem (*) is given. 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