Let $u : \Omega \times [0,T]$ be a function such that $u \in C^{2,1}(\Omega \times [0,T])\cap C^{1}((0,T);L^{2}(\Omega))\cap C([0,T);H_{0}^{1}(\Omega))$ for $\Omega \subset \mathbb{R}$ an unbounded domain.
In order to clarify the meaning of the notation, I will explain some notations mentioned above.
1. $C^{2,1}(\Omega \times [0,T])$ : the function is twice differentiable with respect to spatial domain and once differentiable with respect to time domain.
2. $C^{1}((0,T);L^{2}(\Omega))$ : for any fixed $t \in (0,T)$, $u(\, .\, ,t)\in L^{2}(\Omega)$ and the mapping is once differentiable
3. $C([0,T);H_{0}^{1}(\Omega))$ : for any fixed $t \in [0,T), u(\, .\, ,t)\in H_{0}^{1}(\Omega)$ and the mapping is continuous.
So now I define, a functional $F[\, .\,] = ||\, .\,||_{L^{p}(\Omega)}^{p}$ ($2<p<\infty$) so that I have $F[u(\,.\,)] : [0,T)\to\mathbb{R}$
What I want to show is $F[u(\,.\,)] \in C([0,T);\mathbb{R})$ but I am not sure how to ensure that $||\,.\,||_{L^{p}(\Omega)}$ is finite since $\Omega$ is unbounded and thus I cannot use the embedding of $L^{p}$ to $L^{2}$ for $2<p<\infty$. Furthermore, the dimension $n=1$ so I cannot use any embedding inequality here.
Any help is much appreciated! Thank you!
What you need is the Sobolev embedding $H^1_0(\Omega)\hookrightarrow L^\infty(\Omega)$ (see for example Theorem 8.8 in Functional Analysis, Sobolev Spaces and Partial Differential Equations by Brezis). It implies $u\in C([0,T);L^2(\Omega))\cap C([0,T);L^\infty(\Omega))$. Together with the interpolation inequality $$ \|f\|_p^p\leq \|f\|_2^{2}\|f\|_\infty^{p-2} $$ this yields $u\in C([0,T);L^p(\Omega))$. In particular, $F$ is continuous on $[0,T)$.