# A question about how to use the convexity condition?

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.

• 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 $p''(u_n)$. – Connor Mooney May 10 '19 at 16:21
• @Connor Mooney: Thanks – Hheepp May 10 '19 at 17:21