Let $\Omega \subset \mathbb{R}$ be an unbounded domain and $u \in C([0,\infty);H_{0}^{1}(\Omega))$. Assume there exists a sequence $\{t_{n}\}_{n\in\mathbb{N}}$ such that $t_{n}\to\infty$ and $||u(t_{n})||_{H_{0}^{1}(\Omega)}\to 0$ as $n\to\infty$. Is it possible to show that $||u(t)||_{H_{0}^{1}(\Omega)}\to 0$ as $t\to\infty$? I am not sure how to show this if this is possible. If this is not possible, then is there any counter example? Moreover, can I still obtain similar result if I replace $H_{0}^{1}(\Omega)$ with $X$ a Banach space?

Any hint is much appreciated! Thank you!


Let $u(t)=\sin(t\pi)f$ where $f$ is a fixed non-zero element of $H_0^{1}(\Omega)$. Take $t_n=n$.

  • $\begingroup$ what if I add one more condition such that $\forall n \in \mathbb{N}, u(t_{n})\neq 0$? $\endgroup$ – Evan William Chandra Mar 25 at 8:02
  • 1
    $\begingroup$ Just add $\frac 1 t$ to $\sin(t\pi)$. $\endgroup$ – Kabo Murphy Mar 25 at 8:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.