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In this paper, we first apply parabolic inequalities and a maximum principle to give a new proof for symmetry and monotonicity of solutions to fractional elliptic equations with gradient term by the method of moving planes. Under the condition of suitable initial value, by maximum principles for the fractional parabolic equations, we obtain symmetry and monotonicity of positive solutions for each finite time to nonlinear fractional parabolic equations in a bounded domain and the whole space. More generally, if bounded domain is a ball, then we show that the solution is radially symmetric and monotone decreasing about the origin for each finite time. We firmly believe that parabolic inequalities and a maximum principle introduced here can be conveniently applied to study a variety of nonlocal elliptic and parabolic problems with more general operators and more general nonlinearities.
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참고문헌 (31)
참고문헌 신청
Anatoli V. Babin / 1994 / Symmetrization properties of parabolic equations in symmetric domains / Journal of Dynamics and Differential Equations 6(4):639~658
Xavier Cabré / 2010 / Positive solutions of nonlinear problems involving the square root of the Laplacian / Advances in Mathematics 224(5):2052~2093
Luis Caffarelli / 2007 / An Extension Problem Related to the Fractional Laplacian / Communications in Partial Differential Equations 32(8):1245~1260
Wenxiong Chen / 2018 / Maximum principles for the fractional p-Laplacian and symmetry of solutions / Advances in Mathematics 335:735~758
Wenxiong Chen / 2017 / A direct method of moving planes for the fractional Laplacian / Advances in Mathematics 308:404~437
Xavier Cabré / 2010 / Positive solutions of nonlinear problems involving the square root of the Laplacian / Advances in Mathematics 224(5):2052~2093
Luis Caffarelli / 2007 / An Extension Problem Related to the Fractional Laplacian / Communications in Partial Differential Equations 32(8):1245~1260
Wenxiong Chen / 2018 / Maximum principles for the fractional p-Laplacian and symmetry of solutions / Advances in Mathematics 335:735~758
Wenxiong Chen / 2017 / A direct method of moving planes for the fractional Laplacian / Advances in Mathematics 308:404~437
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