Some nonlinear inequalities and applications

Sufficient conditions are given for the relation ${\lim }_{t\to \mathrm{\infty }}y(t)=0$ to hold, where $y(t)$ is a continuous nonnegative function on $[0,1)$ satisfying some nonlinear inequalities. The results are used for a study of large time behavior of the solutions to nonlinear evolution equations. Example of application is given for a solution to some evolution equation with a nonlinear partial differential operator.