# Is the passage in argument of existence solution of PDE correct?

The passage below is from a fixed point argument in Wang's paper enter link description here pg 566. At the end of the $$I_1$$ calculation, it somehow makes the following estimate $$C\|\phi\|_X (e^{-|x|^2/32t} +w^{-1}(x)) \leq C(p,n,l)\|\phi\|_X w^{-1}(x)$$ assuming according to the author that $$T>1$$. Here $$w^{-1}(x) = (1+|x|)^{-l/(p-1)}$$. My question is whether this passage is really correct? If $$T<1$$ I think it works, because $$e^{-|x|^2/32t}\leq e^{-|x|^2/32}$$ so using that $$w(x)e^{-|x|^2/32}, $$C_1$$ constant, $$(e^{-|x|^2/32t} +w^{-1}(x)) \leq (w^{-1}(x)w(x)e^{-|x|^2/32} +w^{-1}(x))$$

But the same passage for $$T>1$$ I can't see how it can be done. Is it something obvious? I appreciate any clarifications.

This is probably a typo: probably the author meant $$T<1$$ here, instead of $$T>1$$. A bit later, in lines 3- and 2- at the bottom of the same page, it is explicitly assumed that $$T$$ is $$<1$$ and furthermore small enough to make the standard contraction argument work.

It is a typo.

Look at the bottom of the same page: where the author says to choose "$$T = T(p,n,l,\|\phi\|_X) < 1$$ such that ..."

The author is proving local existence here, and it makes no sense to assume a minimum time of existence.

• Two almost identical answers, but obviously independent of each other, as one was posted seconds after the other! :-) Dec 7, 2023 at 6:14
• Looks like you beat me by a tiny bit according to the timestamps. :-) Dec 7, 2023 at 6:45
• I'm grateful for both comments. In fact, I was confused, because when we read a work we tend to trust what the author writes, so I thought it was something I wasn't seeing. Dec 7, 2023 at 9:56
• @Ilovemath I assume you know this already, but always be ready for typos, mistakes and wrong arguments, whether or not the work is published or ''trustworthy''. Dec 7, 2023 at 14:09
• @HollisWilliams In general, I'm very careful about trying to understand all the details when reading a paper, but as this work is quite famous, I believed it could really be something that I wasn't calculating correctly. But the lesson remains! Dec 8, 2023 at 2:20