Can anyone see how (1) lead to (2)?
\begin{align}
\int_{\Omega}\{-\Delta(\phi_{\varepsilon}(b^{\varepsilon}))(b^{\varepsilon})^p\}(t)&=\int_{\Omega}\{-\Delta((\varepsilon+b^{\varepsilon})^k-{\varepsilon})^k)(b^{\varepsilon})^p\}(t)\quad (1)\\
&=\int_{\Omega}\{kp(\varepsilon+b^{\varepsilon})^{k-1}(b^{\varepsilon})^{p-1}|\nabla b^{\varepsilon}|^2\}\quad (2)
\end{align}
Here
$$
\phi_{\varepsilon}(s):=(s+\varepsilon)^k-{\varepsilon}^k
$$
$$
\nabla\phi_{\varepsilon}(b)=0
$$
and $\varepsilon$ is a small positive number.
$b\in C(\bar{Q}_T)$, where $Q_T=\Omega\times (0,T]$
Note: The Laplacian applies only to $\phi_{\varepsilon}(b^{\varepsilon})$, not $(b^{\varepsilon})^p$ .