Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse.

Does there exist a solution $u \in L^2(0,T;H^1(\Omega))$ with $\gamma(u) \in W(0,T)$ such that $$\langle (\gamma u)_t, \gamma(v)\rangle + \int_\Omega \nabla u \nabla v = 0$$ for all $v \in L^2(0,T;H^1(\Omega))$?

Is there some literature about such problems? I particularly am interested in variational/energy/Galerkin methods.

The problem is that the trace mapping is not invertible, so the converting the problem to $$\int_{\partial\Omega} z_t w+ \int_\Omega \nabla \xi(u) \nabla \xi(w) = 0$$ for all $w \in L^2(0,T;H^{\frac 12}(\Omega))$ does not help.