# Meaning of Compactness

Let $$\Omega \subset\mathbb{R}$$ be a bounded domain (interval) and observe the following problem : \begin{align*} (P) \begin{cases} u_{t} = \Delta u + |u|^{p-1}u\, \quad x\in\Omega, t>0\\ u(0,x) = u_{0}(x), \quad x\in\Omega\\ u(t,x)|_{\partial\Omega} = 0, \quad \forall t\geq0 \end{cases} \end{align*}

It is well-known that $$\forall u_{0} \in H_{0}^{1}(\Omega), \, \exists T_{m}>0$$ such that $$\exists! u \in C([0,T_{m});H_{0}^{1}(\Omega))\cap C^{1}((0,T_{m});L^{2}(\Omega))$$ satisfies $$(P)$$. Moreover, $$u$$ becomes a classical solution of $$(P)$$ for any $$t>0$$.

Now, my problem is that I want to show that by assuming $$T_{m} = \infty$$ then $$\forall t \in [0,\infty), \, u(t)$$ is contained in a compact set in $$H_{0}^{1}(\Omega)$$.

Basically, I want to apply a Theorem 4.3.3 from the book "Geometric Theory of Semilinear Parabolic Equations" written by Dan Henry.

This is the statement of Theorem 4.3.3 from the book :
Let $$\{S(t),t\geq0\}$$ be a dynamical system lies in a compact set in complete metric space $$C$$. Then, $$\omega(x_{0})$$ is nonempty, compact, invariant, and connected, and $$\text{dist}(S(t)x_{0},\omega(x_{0}))\to 0$$ as $$t\to\infty$$

For convenience, I will mention the definition of dynamical system and $$\omega(x_{0})$$.
Definition (Dynamical System)
A dynamical system (nonlinear semigroup) on $$C$$ is a family of maps $$\{S(t) : C\to C, \, t\geq 0\}$$ such that
(i) $$\forall t\geq 0, \, S(t)$$ is continuous from $$C$$ to $$C$$.
(ii) $$\forall x \in C$$, $$t \mapsto S(t)x$$ is continuous.
(iii) $$S(0) = I$$, an identity mapping on $$C$$.
(iv) $$\forall t,\tau\geq0, \,S(t)(S(\tau)x) = S(t+\tau)x$$

Definition Let $$x_{0} \in C, \gamma(x_{0}) = \{S(t)x_{0}; t \geq 0\}$$ is the orbit through $$x_{0}$$, then $$\omega(x_{0}) := \{x \in C\, |\,\exists\{t_{n}\}_{n\in\mathbb{N}} \text{ with } t_{n}\to\infty \text{ s.t. } S(t_{n})x_{0}\to x \text{ as }n\to\infty\}$$

So, first I take $$C = H_{0}^{1}(\Omega)$$ and define $$S(t)u_{0} = u(t)$$. Now, my problems in summary are :
1. Can I really define $$S(t)$$ in that way and assure that (iv) holds? How to show (iv) holds?
2. Again, how to show that $$\{u(t),t\geq0\}$$ is contained in a compact set in $$H_{0}^{1}(\Omega)$$?

Any help is much appreciated. I am actually trying to read some results in a paper written by Ryo Ikehata anda Takashi Suzuki with the title "Stable and unstable sets for evolution equations of parabolic and hyperbolic type" published in Hiroshima Math Journal 26 (1996), 475-491.