Maximum principle for the fractional N-Laplacian flow

Abstract We deal with a family of the fractional N-Laplacian heat flows with the variable exponent time-derivative on the Orlicz-Sobolev spaces. We get the maximum principle for these problems. We use the approximating method to get this result: We first show existence of a unique family of the approximating weak solutions from the variable exponent difference fractional N-Laplacian problems. We next show the maximum principle for the family of the approximating weak solution from the variable exponent difference fractional N-Laplacian problem, show the convergence of a family of the approximating weak solutions to the limits, and then obtain the maximum principle for the weak solution of a family of the fractional N-Laplacian heat flows with the variable exponent time-derivative on the Orlicz-Sobolev spaces. AMS 2000 Mathematics Subject Classifications: 35A01, 35A25, 35D30, 35B50, 35K35, 46E30