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!