  We prove existence of variational solutions for a class of nonlocal evolution
equations whose prototype is the double phase equation \begin{align*}
\partial_t u &+ \text{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}}\\&+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qr}}
\,dy = 0. \end{align*} The approach of minimization of parameter-dependent
convex functionals over space-time trajectories requires only appropriate
convexity and coercivity assumptions on the nonlocal operator. As the parameter
tends to zero, we recover variational solutions. Under further growth
conditions, these variational solutions are global weak solutions. Further,
this provides a direct minimization approach to approximation of nonlocal
evolution equations.

Analysis of PDEs