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In the case $\Omega=\mathbb{R}^n$ we have \begin{equation} L^2(0,T;W^{1,2}(\Omega))\cap W^{1,2}(0,T;W^{-1,2}(\Omega))\hookrightarrow BUC([0,T];X) \end{equation} where $X$ is given via real interpolation: \begin{equation} X=\Big(W^{1,2}(\Omega)),W^{-1,2}(\Omega))\Big)_{1/2,2}=B^0_{2,2}(\Omega). \end{equation} This is basically contained in Linear and quasilinear parabolic problems I by H. Amann (Theorem III.4.10.2). Since $B^0_{2,2}(\Omega)=L^2(\Omega)$ the desired result follows. Now the case of smooth bounded $\Omega$ should follow via extension and restriction.