Compactness of Sobolev Space

Let $$\Omega \subset \mathbb{R}$$ be a bounded domain and $$2. Assume $$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!

• 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. – lsir Mar 26 at 2:25
• I see, thank you for your comment! It seems to help a bit for my problem – Evan William Chandra Mar 26 at 3:16