Existence of variational solutions to doubly nonlinear nonlocal evolution equations via minimizing movements

We prove existence of variational solutions for a class doubly nonlinear nonlocal evolution equations whose prototype is the double phase equation $$\begin{aligned}&\partial _t u^m + P.V.\int _{\mathbb {R}^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\&\qquad \quad +a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qr}} \,\mathrm{d}y = 0,\,m>0,\,p>1,\,s,r\in (0,1). \end{aligned}$$ make use approach minimizing movements pioneered by De Giorgi (in Boundary value problems partial differential and applications, volume 29 RMA Res. Notes Appl. Math., Masson, Paris, pp 81–98, 1993. https://mathscinet.ams.org/mathscinet-getitem?mr=1260440 ) Ambrosio (Accademia Nazionale delle Scienze detta dei XL. Rendiconti. Serie V. Memorie di Matematica e Applicazioni. Parte I, 19:191–246) refined Bögelein, Duzaar, Marcellini coauthors to study parabolic with nonstandard growth.