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At page 5 (125), seven line after the Proof of Theorem 2.2(i), of the following article.

THE HEAT EQUATION WITH A SINGULAR POTENTIAL

the authors say that since $p$ is convex, we can deduce that

$$ \int_{\delta}^{t} \int_{\Omega} p(u_n) (- \Delta \phi) \leq \int_{\delta}^{t} \int_{\Omega} \nabla u_n \cdot \nabla (\phi \, p'(u_n)). $$

I am confused about how convexity is used. I will be thanked if anyone could help me.

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    $\begingroup$ If $\phi$ is a positive cutoff, then after integrating by parts on the left side (in space) we get the right side, minus an integral of positive quantities times $p''(u_n)$. $\endgroup$ – Connor Mooney May 10 '19 at 16:21
  • $\begingroup$ @Connor Mooney: Thanks $\endgroup$ – Hheepp May 10 '19 at 17:21

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