  We study the solutions of the stochastic heat equation with multiplicative
space-time white noise. We prove a comparison theorem between the solutions of
stochastic heat equations with the same noise coefficient which is H\"{o}lder
continuous of index $\gamma>3/4$, and drift coefficients that are Lipschitz
continuous. Later we use the comparison theorem to get sufficient conditions
for the pathwise uniqueness for solutions of the stochastic heat equation, when
both the white noise and the drift coefficients are H\"{o}lder continuous.
