Uniform $L^{2}$-decay of solutions for the linear 

heat equations will be given. In order to derive 

the $L^{2}$-decay of solutions, the modified method 

of Morawetz~\cite{mora} will be used and we shall show 

that the $L^{2}$ norm of solutions decays like 

$O(t^{-1})$ as $t \to +\infty$ for some kinds of 

weighted initial data. Furthermore, by the same argument, 

one can also derive the $L^{2}$-bound and $L^{2}$-decay 

for weak solutions of the linear free and dissipative 

wave equations, respectively.