For a bounded smooth domain $\Omega$ in $\mathbb{R}^N$, We know that if $u\in L^2(0,T;H_0^1(\Omega)$ and $\partial_tu\in L^2(0,T;H^{-1}(\Omega))$ then $u\in C^0([0,T];L^2(\Omega))$.
However, now we only have $u\in L^{\infty}(0,T;L^2(\Omega))\cap L^2(0,T;H_0^1(\Omega)$ and $\partial_tu\in L^2(0,T;H^{-1}(\Omega))+L^1(0,T;L^1(\Omega))$. Is $u\in C^0([0,T];L^2(\Omega))$?
In paper "Geredeli, Pelin G.; Khanmamedov, Azer. Long-time dynamics of the parabolic $p$-Laplacian equation. Commun. Pure Appl. Anal. 12 (2013), no. 2, 735--754" I do not know why they obtain the conclude statement under (3.18)?
Please help me!