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.

Automated summary

We prove the local boundedness of variational solutions to the double phase equation under certain restrictions on s, s', p, q, and the non-negative function a(x, y).