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Let $\Omega \subset \mathbb{R}$ be a bounded domain and $2<p<\infty$. Assume $u \in C^{2,1}(\Omega \times (0,\infty))\cap C^{1}((0,\infty);L^{2}(\Omega))\cap C([0,\infty);H_{0}^{1}(\Omega))$ such that the functional $J[u(\,\cdot\,)]:= \frac{1}{2}||u(\,\cdot\,)||_{H_{0}^{1}(\Omega)^{2}}^{2} - \frac{1}{p}||u(\,\cdot\,)||_{p}^{p}$ is non-increasing functional over time $t$. Then, suppose that there exists a time sequence $\{t_{n}\}_{n\in\mathbb{N}}\subset [0,\infty)$ such that $t_{n}\to\infty$ such that $J[u(t_{n})]\to 0$ as $n\to\infty$. Finally, we define $u_{n} := u(t_{n})$ in $H_{0}^{1}(\Omega)$. So, my question is that "is it possible to find $v\in H_{0}^{1}(\Omega)$ such that $u_{n}\to v$ in $H_{0}^{1}(\Omega)$"? As far as I know, $H_{0}^{1}(\Omega)$ is not a compact space and thus any bounded sequence might not have a convergent subsequence but can we show that it is compact in the case for 1 dimensional bounded domain?

Any help will be much appreciated!

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    $\begingroup$ In general you cannot show that $H^1_0$ is compact since that would imply that the space is finite dimensional. Given a bounded sequence in $H^1_0$ you can extract a subsequence converging in $L^2$ due to the compact embedding $H^1_0 \subseteq L^2$ (see Rellich–Kondrachov theorem). But anyway you have many additional assumptions in your problem so my comment is only on your compactness question for $H^1_0$ not on the possibility of finding the desired solution. $\endgroup$ – lsir Mar 26 at 2:25
  • $\begingroup$ I see, thank you for your comment! It seems to help a bit for my problem $\endgroup$ – Evan William Chandra Mar 26 at 3:16

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