  This is a study of a class of nonlocal nonlinear diffusion equations. We
present a strong maximum principle for nonlocal time-dependent Dirichlet
problems. Results are for bounded functions of space, rather than
(semi)-continuous functions. Solutions that attain interior global extrema must
be identically trivial. However, depending on the nonlinearity, trivial
solutions may not be constant in space; they may have an infinite number of
discontinuities, for example. We give examples of nonconstant trivial solutions
for different nonlinearities. For porous medium-type equations, these functions
do not solve the associated classical differential equations, even those in
weak form. We also show that these problems are globally wellposed for
Lipschitz, nonnegative diffusion coefficients.
