The first and second initial-boundary value problems for second order sem-ilinear parabolic equation△u+u

t+λρ(x

t)f(u)=0are considered.The maximum principles of parabolic equations are used todeduce that certain functionals defined for solutions of the problems attain amaximum at a point at which |▽_xu|=0.This result is then used to determinebounds for solutions of a class of semilinear parabolic equations.