On the speed of decay of solutions to some partial differential equations

In this paper we show that it is possible to estimate the speed of decay (as t→+∞) of a function u(t) simply by proving that it satisfies certain integral inequalities. Moreover, we study the relationship between these integral inequalities and the phenomenon of the extinction in finite time proving that the extinction appears for a suitable choice of the exponents in these inequalities. Finally, we give same easy example of possible applications of these general results to the description of the asymptotic behavior of the solutions to some different evolution problems.