Parabolic double phase obstacle problems

We prove existence results for the parabolic double phase obstacle problem: Find $u\in K\subset X_0$ with $u(\cdot ,0)=0$ satisfying $0\in u_t+Au+F\left(u\right)+\partial I_K\left(u\right)\text{in}{X}_0^{\ast },$where $A:X_0\to X_0^{\ast }$ given by $Au≔-div\left({|\nabla u|}^{p-2}\nabla u+\mu \left(x\right){|\nabla u|}^{q-2}\nabla u\right)\text{for}u\in X_0,$is the double phase operator acting on $X_0=L^p(0,\tau ;W_0^{1,H}\left(\Omega \right))$ with $W_0^{1,H}\left(\Omega \right)$ denoting the associated Musielak–Orlicz Sobolev space with generalized homogeneous boundary values. The obstacle is represented by the closed convex set $K$ with the obstacle function $\psi$ through $K=\{v\in X_0:v(x,t)\le \psi (x,t)\text{for a.a.}(x,t)\in Q=\Omega \times (0,\tau )\}$and $I_K$ is the indicator function related to $K$ with $\partial I_K$ denoting its subdifferential in the sense of convex analysis.