Local boundedness of variational solutions to nonlocal double phase parabolic equations

We prove local boundedness of variational solutions to the double phase equation partial derivative(t)u+ P.V. integral(N)(R) |u(x, t) - u(y, t)|(p-2)( u(x, t)- u(y, t)) |x - y|(N+ ps) + a(x, y) | u(x, t) - u(y, t)|(q-2)( u(x, t)- u(y, t)) | x - y|(N+ qs') similar to dy = 0, under the restrictions s, s'is an element of(0, 1), 1 < p <= q <= p(2s+ N/)N and the non-negative function (x, y) proves. a(x, y) is assumed to be measurable and bounded.