Does weak convergence in $L^\infty(\mathbb{R},L^p(\mathbb{R}^n))$ imply strong convergence in $L^p_{loc}(\mathbb{R}\times\mathbb{R}^n)$?

Question

Does weak convergence in $L^\infty\left(\mathbb{R},L^p\left(\mathbb{R}^n\right)\right)$ imply strong convergence in $L^p_{loc}\left(\mathbb{R}\times\mathbb{R}^n\right)$ up to a subsequence ?