Let $H^s(\mathbb{R}^n)$ denotes the fractional Sobolev space consists of those elements $f:\mathbb{R^n}\to \mathbb{R}$ such that $$ ||f||^{2}_{L^2}(\mathbb{R^n})+\int_{\mathbb{R^n}}\int_{\mathbb{R^n}}\frac{|f(x)-f(y)|^2}{|x-y|^{n+2s}}\,dx dy<\infty. $$ The space $L^2(t_1,t_2;H^s(\mathbb{R}^n))$ consists of those $v:\mathbb{R^n}\times(t_1.t_2)\to\mathbb{R}$ such that $$ \int_{t_1}^{t_2}\int_{\mathbb{R^n}}|v|^2 dx dt+\int_{t_1}^{t_2}\int_{\mathbb{R^n}}\int_{\mathbb{R^n}}\frac{|v(x,t)-v(y,t)|^2}{|x-y|^{n+2s}}\,dx dy dt<\infty. $$ Let $u\in H^s(\mathbb{R}^n)$ be a bounded function. Let $\psi:\mathbb{R^n}\to\mathbb{R}$ be smooth and $\eta:\mathbb{R}\to\mathbb{R}$ be smooth with compact support. Then if $p>1$, is the function $\phi(x,t)=u^p\psi(x)\eta(t)\in L^2(t_1,t_2;H^s(\mathbb{R}^n))$?
Can anybody give some hint. Thanks.