In this paper, we are devoted to studying the following nonlocal elliptic-parabolic equations involving the fractional (p, 2)-Laplacian \(\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t^\beta u+ (-\Delta )_{p}^\alpha u+(-\Delta )_{2,a}^{\iota }u=\lambda |u|^{q-2}uv+g(x,t) & \text{ in } \Omega \times \mathbb {R}^{+},\\ (-\Delta )^\gamma v=|u|^{q} & \text{ in } \Omega \times \mathbb {R}^{+},\\ u(x,t)=v(x,t)=0\ \ & \text{ in } (\mathbb {R}^N\setminus \Omega )\times \mathbb {R}^+,\\ u(x,0)=u_0(x)\ & \text{ in } \Omega ,\\ \end{array}\right. } \end{aligned}\) where \(\Omega \subset \mathbb {R}^N\) is a bounded domain with Lipschitz boundary, \((-\Delta )_{p}^{\alpha }+(-\Delta )_{2,a}^{\iota }\) is the fractional (p, 2)-Laplacian with \(0<{\iota }<\alpha <1\) , \(p,q\ge 2\) , \(a:\mathbb {R}^N\times \mathbb {R}^N\rightarrow [0,\infty )\) is a bounded function, \(\partial _t^{\beta }\) is the Riemann-Liouville time fractional derivative with \(0<\beta <1\) , \(\lambda \) is a parameter, and \(g\in L^\infty (0,\infty ;L^2(\Omega ))\) . The existence theory of solutions is established by applying the Galerkin method combined with fractional calculus theory. Then, by the comparison theorem, the uniqueness of the global weak solution is derived. Moreover, under some suitable assumptions, we also give a decay estimate of solutions. There are two main features of this paper. First, our problem is the combination of both the Riemann-Liouville time fractional derivative and the fractional (p, 2)-Laplacian operator. In particular, the fractional Laplacian \((-\Delta )_{2,a}^{\iota }\) has a weight function \(a(\cdot )\) which plays a role in transforming between two states. If \(p\ne 2\) , the presence of two fractional operators with different growth, which generates a double phase anisotropic energy. Second, by the Lax-Milgram theorem, the above problem presents a Choquard nonlinear term, which also leads to non-local characteristics.
