Often regarding pde's I see that a weak solution has to be in the space $L^p_{\text{loc}}(0,T,W^{1,p}_{\text{loc}})$ on some domain. What exactly does this mean? I understand that the function and it's weak gradient need to be in $L^p$, but what about that "first" $L^p$ term?
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Are you referring to the $L^p_{loc}$? A function $f$ is said to be locally $L^p$ if $f^p$ is integrable on bounded sets. – Nick F Dec 16 '22 at 18:06
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Thanks! No I didn't mean that. I don't understand what that $L^p_{\text{loc}}$ term does there. If there only was $u\in W^{1,p}_{\text{loc}}$ I would understand it. – HelloEveryone Dec 16 '22 at 18:15
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Is it meant to be $L^p_{\mathrm loc}$ as a function of time, taking values in $W^{1,p}_{\mathrm loc}$ in the spatial coordinate? – paul garrett Dec 16 '22 at 18:40
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@paulgarrett Now i understood it, thank's! It indeed is as you said. – HelloEveryone Dec 16 '22 at 20:42