Let $\Omega\subset\mathbb{R}$ be a bounded interval, $\{u_{n}(t)\}_{n\in\mathbb{N}}$ be a bounded sequence in $H_{0}^{1}(\Omega)$ and define $J[u_{n}(t)] = \frac{1}{2}||u_{n}(t)||_{H_{0}^{1}(\Omega)}^{2}-\frac{1}{p}||u_{n}(t)||_{p}^{p}$ such that $\frac{d}{dt}J[u_{n}(t_{1})] \to 0$ as $n\to\infty$ for $t_{1} \in [0,\infty)$ and $2<p<\infty$.
By the property of Hilbert Space, we can have a subsequence $\{u_{n_{k}}(t_{1})\}_{k\in\mathbb{N}}$ such that $u_{n_{k}(t_{1})}\to v$ weakly in $H_{0}^{1}(\Omega)$.
Let $J'[u_{n}(t_{1})]$ be the Frechet derivative of $J[u_{n}(t_{1})]$. Can I obtain $J'[u_{n}(t_{1})]\to 0$ as $n\to\infty$?
For your information, the functional $J[\,\cdot\,]$ is related to the following semilinear pde ($2<p<\infty$) \begin{equation*} \begin{cases} u_{t} = \Delta u-u+u|u|^{p-2} &\text{ in }\Omega\times(0,\infty)\\ u(\cdot,0) = u_{0} &\text{ in }\Omega\\ u(0,t) = u(b,t)= 0 &\text{ in }(0,\infty) \end{cases} \end{equation*}
I am trying to obtain strong convergence in $H_{0}^{1}(\Omega)$ from weak convergence but I need $J'[u_{n}(t_{1})]\to 0$ as $n\to\infty$ in order to do it.
Any help is much appreciated! Thank you very much!