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March 2013, 12(2): 973-984. doi: 10.3934/cpaa.2013.12.973
Decay of solutions to fractal parabolic conservation laws with large initial data
Fengbai Li 1, and Feng Rong 1,
Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, China, China
Received October 2011 Revised February 2012 Published September 2012
In this paper, we study the time-asymptotic behavior of solutions to the Cauchy problem for multi-dimensional parabolic conservation laws with fractional dissipation. For arbitrarily large initial data, we obtain the optimal decay rates in $L^2$ and homogeneous Sobolev spaces for solutions to the equation with the power of Laplacian $\frac{1}{2} < \alpha \le 1$ by using the time-frequency decomposition method and the energy method. The argument is based on a maximum principle.
Keywords: optimal decay rate., parabolic conservation laws, large initial data, Time-frequency cut-off operator.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35R1.
Citation: Fengbai Li, Feng Rong. Decay of solutions to fractal parabolic conservation laws with large initial data. Communications on Pure & Applied Analysis, 2013, 12 (2) : 973-984. doi: 10.3934/cpaa.2013.12.973
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Fengbai Li Feng Rong