  White-noise case stochastic heat equation was derived from Dawson-Watanabe
superprocess. The pathwise uniqueness of their solutions with non-Lipschitz
coefficients has attracted wide attention and in particular, the square-root
coefficient case was listed as a long standing open problem in the famous
literature (i.e. Perkins \cite{Perkins02} p217.) This short note gives an
affirmative answer to this open problem.
  Our idea is using the $\varphi_k(x)$ function constructed by Yamada-Watanable
to prove that
  \beqlb \lab{key}
  \E_{\mu}| X_{t}(z)-Y_{t}(z) | = \lim_{k \ra \infty}
\E_{\mu}|\int_{\r}\frac{1}{\sqrt{\epsilon_k}} \rho_{\epsilon_k}(z-x)[
X_{t}(x)-Y_{t}(x)]dx |=0 .
  \eeqlb
  However, $\frac{1}{\sqrt{\epsilon_k}}
\rho_{\epsilon_k}(x)=p_{\epsilon_k}(x)$, the stable kernel, is not
square-integrable uniformly in $k$ and we cannot directly use the heat kernel
or stable kernel convolution transformation. This forces us to go a new way in
which we have used the linear and reciprocal linear properties of $\varphi_k$
and $\varphi^{\prime \prime}_k$, respectively, making a $\rho_{\epsilon_k}$
convolution transformation first, then, taking expectation and multiplying
singularity factor $\frac{1}{\sqrt{\epsilon_k}}$ and finally letting $k \ra
\infty$ to estimate and derive (\ref{key}). In this way, we have avoided the
non-uniform square integrability of the stochastic integral terms which involve
the singularity. The same idea also can be used to prove the pathwise
uniqueness of the nonnegative solutions of stochastic heat equation with
$\alpha$ coefficients ($1/2 \leq \alpha$) since by a same idea we can construct
the $\varphi_k(x)$ functions with $1/2 \leq \alpha$.
