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

A strong maximum principle for nonlinear nonlocal diffusion equations