For an evolution (time-dependent) problem with a Cauchy data, you must consider the pair $W:=(w,\partial_tw)$, which satisfies a first-order equation in the time variable. According to your a priori estimate, a convenient space is $$L^2(0,T;H^2(\Omega)\times L^2(0,T;L^2(\Omega),$$ with $\Omega=(-\ell_1,\ell_1)\times(-\ell_2,\ell_2)$ your spatial domain. However, because of your Dirichlet boundary condition, it is better to consider $$L^2(0,T;(H^2\cap H^1_0)(\Omega)\times L^2(0,T;L^2(\Omega).$$ If you are willing to employ the Hille-Yosida Theorem in semi-group theorey, the second-order boundary condition will come in the domain of the operator.

For an evolution (time-dependent) problem with a Cauchy data, you must consider the pair $W:=(w,\partial_tw)$, which satisfies a first-order equation in the time variable. According to your a priori estimate, a convenient space is $$ L^2(0,T;H^2(\Omega))\times L^2(0,T;L^2(\Omega)), $$ with $\Omega=(-\ell_1,\ell_1)\times(-\ell_2,\ell_2)$ your spatial domain. However, because of your Dirichlet boundary condition, it is better to consider $$ L^2(0,T;(H^2\cap H^1_0)(\Omega))\times L^2(0,T;L^2(\Omega)). $$ If you are willing to employ the Hille-Yosida Theorem in semi-group theory, the second-order boundary condition will come in the domain of the operator.