from we infer the bound $$\label{eq:cacc_iv}
    J_{1}
    +
    J_{21}
    +
    J_{22}
    \le
    J_{3}
    +
    J_{23}
    \,,$$ where $$\begin{aligned}
    J_{21}
    &=
      \gamma_{0}
      \int_{0}^{t}
      \sum_{x,y\in V}
      {\left|
      {D}_{y}
      {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\frac{p+\theta-1}{p}}
      \right|}^{p}
      \zeta(y)^{p}
      {\omega}(x,y)
      \eta(\tau)^{p}
      {\,\textup{\textmd{d}}}\tau
      \,,
    \\
    J_{22}
    &=
      \frac{1}{4}
      \int_{0}^{t}
      \sum_{x,y\in V}
      {\lvert{D}_{y}{u}(x,\tau)\rvert}^{p-2}
      {D}_{y}{u}(x,\tau)
      {D}_{y}[
      {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\theta}
      ]
      \zeta(y)^{p}
      {\omega}(x,y)
      \eta(\tau)^{p}
      {\,\textup{\textmd{d}}}\tau
      \,,
    \\
    J_{23}
    &=
      \frac{1}{2}
      \int_{0}^{t}
      \sum_{x,y\in V}
      {\lvert{D}_{y}{u}(x,\tau)\rvert}^{p-1}
      {\lvert{D}_{y}\zeta(x)^{p}\rvert}
      {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\theta}
      \eta(\tau)^{p}
      {\omega}(x,y)
      {\,\textup{\textmd{d}}}\tau
      \,.
  \end{aligned}$$ The reason to preserve the fraction $J_{22}$ of $J_{2}$ (rather than treating it as in $J_{21}$) will become apparent presently. Let us introduce the functions $$\begin{gathered}
    H(x,y;r)
    =
    \max[
    {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-r)_{+}}{\left({u}(x,\tau)-r\right)_{+}}}}
    ,
    {{\ifthenelse{\equal{*}{*}}{({u}(y,\tau)-r)_{+}}{\left({u}(y,\tau)-r\right)_{+}}}}
    ]
    \,,
    \\
    \chi_{x,y}
    =
    1
    \,,
    \quad
    \text{if $H(x,y;h)>0$;}
    \qquad
    \chi_{x,y}
    =
    0
    \,,
    \quad
    \text{if $H(x,y;h)=0$.}
  \end{gathered}$$ Note that $r>0$ is arbitrary in the definition of $H$ but we fix $r=h$ in the definition of $\chi_{x,y}$. Next we select $0<k<h$; by elementary calculations and Young’s inequality we get $$\begin{split}
      J_{23}
      &\le
      \frac{p}{2}
      \int_{0}^{t}
      \sum_{x,y\in V}
      {\lvert{D}_{y} {u}(x,\tau)\rvert}^{p-1}
      {\lvert{D}_{y} \zeta(x)\rvert}
      (\zeta(x)+\zeta(y))^{p-1}
      H(x,y;k)^{\theta}
      \chi_{x,y}
      {\omega}(x,y)
      \eta(\tau)^{p}
      {\,\textup{\textmd{d}}}\tau
      \\
      &\le
      {\varepsilon}\int_{0}^{t}
      \sum_{x,y\in V}
      {\lvert{D}_{y}{u}(x,\tau)\rvert}^{p}
      (\zeta(x)^{p}+\zeta(y)^{p})
      H(x,y;k)^{\theta-1}
      \chi_{x,y}
      {\omega}(x,y)
      \eta(\tau)^{p}
      {\,\textup{\textmd{d}}}\tau
      \\
      &\quad+
      \gamma {\varepsilon}^{1-p}
      \int_{0}^{t}
      \sum_{x,y\in V}
      {\lvert{D}_{y}\zeta(x)\rvert}^{p}
      H(x,y;k)^{p+\theta-1}
      {\omega}(x,y)
      \eta(\tau)^{p}
      {\,\textup{\textmd{d}}}\tau
      =:
      J_{231}
      +
      J_{232}
      \,.
    \end{split}$$ We want to absorb partially the 