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May 17 at 19:33 comment added Iosif Pinelis @Theleb : Thank you for your comment. The typo is now fixed. I agree with what you said about your example.
May 17 at 19:30 history edited Iosif Pinelis CC BY-SA 4.0
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May 17 at 14:32 comment added Theleb OK. There is a typo : $t^\frac 23$ --> $t^2$. Your answer provides a counter-example where $\phi$ is continuous. I posted another answer which (in my opinion) is less involved computationally (since $\phi$ is an indicator function there).
May 17 at 12:19 comment added Iosif Pinelis @Theleb : I don't understand your comment. However, if you are satisfied with this answer, these guidelines may be relevant now.
May 17 at 8:01 comment added Theleb Thanks for your complete answer ! Afterwards, I also thought of a counter-example. Take a $phi(t)$ as the indicator function of a set of size $t^2$ inside $\Omega$, the differential inequality is verified while starting at $0$...
May 17 at 1:52 history edited Iosif Pinelis CC BY-SA 4.0
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May 17 at 1:41 history undeleted Iosif Pinelis
May 17 at 1:41 history edited Iosif Pinelis CC BY-SA 4.0
added 4 characters in body
May 17 at 1:27 history deleted Iosif Pinelis via Vote
May 16 at 21:43 history answered Iosif Pinelis CC BY-SA 4.0