# American Institute of Mathematical Sciences

February  2016, 36(2): 1041-1060. doi: 10.3934/dcds.2016.36.1041

## Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy

 1 School of Mathematics, Jilin University, Changchun 130012

Received  June 2014 Revised  January 2015 Published  August 2015

This paper concerns the boundary behavior and the asymptotic behavior of solutions to a class of boundary-initial parabolic problems with boundary degeneracy. At the degenerate boundary, it is shown that the diffusion vanishes and the solution possesses the invariability if the degeneracy is sufficiently strong. As to the asymptotic behavior, it is proved that the decay rate is an exponential function if the degeneracy is weak enough, while a power function if it is not.
Citation: Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041
##### References:
 [1] F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability,, Journal of Evolution Equations, 6 (2006), 161.  doi: 10.1007/s00028-006-0222-6.  Google Scholar [2] F. Black and M. Scholes, The pricing of options and corporate liabilities,, Journal of Political Economy, 81 (1973), 637.  doi: 10.1086/260062.  Google Scholar [3] P. Cannarsa, G. Fragnelli and J. Vancostenoble, Linear degenerate parabolic equations in bounded domains: Controllability and observability,, in Systems, 202 (2006), 163.  doi: 10.1007/0-387-33882-9_15.  Google Scholar [4] P. Cannarsa, G. Fragnelli and J. Vancostenoble, Regional controllability of semilinear degenerate parabolic equations in bounded domains,, Journal of Mathematical Analysis and Applications, 320 (2006), 804.  doi: 10.1016/j.jmaa.2005.07.006.  Google Scholar [5] P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations,, Communications on Pure and Applied Analysis, 3 (2004), 607.  doi: 10.3934/cpaa.2004.3.607.  Google Scholar [6] P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations,, Advances in Differential Equations, 10 (2005), 153.   Google Scholar [7] P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators,, SIAM Journal on Control and Optimization, 47 (2008), 1.  doi: 10.1137/04062062X.  Google Scholar [8] L. C. Evans, Partial Differential Equations,, $1^{st}$ edition, (1998).   Google Scholar [9] P. Martinez, J. P. Raymond and J. Vancostenoble, Regional null controllability of a linearized Crocco-type equation,, SIAM Journal on Control and Optimization, 42 (2003), 709.  doi: 10.1137/S0363012902403547.  Google Scholar [10] P. Martinez, J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations,, Journal of Evolution Equations, 6 (2006), 325.  doi: 10.1007/s00028-006-0214-6.  Google Scholar [11] G. R. North, L. Howard, D. Pollard and B. Wielicki, Variational formulation of Budyko-Sellers climate models,, Journal of the Atmospheric Sciences, 36 (1979), 255.  doi: 10.1175/1520-0469(1979)036<0255:VFOBSC>2.0.CO;2.  Google Scholar [12] O. A. Ole\uinik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form,, American Mathematical Society, (1973).   Google Scholar [13] C. Wang, Approximate controllability of a class of degenerate systems,, Applied Mathematics and Computation, 203 (2008), 447.  doi: 10.1016/j.amc.2008.04.056.  Google Scholar [14] C. Wang, Approximate controllability of a class of semilinear systems with boundary degeneracy,, Journal of Evolution Equations, 10 (2010), 163.  doi: 10.1007/s00028-009-0044-4.  Google Scholar [15] J. Yin and C. Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy,, Journal of Differential Equations, 237 (2007), 421.  doi: 10.1016/j.jde.2007.03.012.  Google Scholar

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##### References:
 [1] F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability,, Journal of Evolution Equations, 6 (2006), 161.  doi: 10.1007/s00028-006-0222-6.  Google Scholar [2] F. Black and M. Scholes, The pricing of options and corporate liabilities,, Journal of Political Economy, 81 (1973), 637.  doi: 10.1086/260062.  Google Scholar [3] P. Cannarsa, G. Fragnelli and J. Vancostenoble, Linear degenerate parabolic equations in bounded domains: Controllability and observability,, in Systems, 202 (2006), 163.  doi: 10.1007/0-387-33882-9_15.  Google Scholar [4] P. Cannarsa, G. Fragnelli and J. Vancostenoble, Regional controllability of semilinear degenerate parabolic equations in bounded domains,, Journal of Mathematical Analysis and Applications, 320 (2006), 804.  doi: 10.1016/j.jmaa.2005.07.006.  Google Scholar [5] P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations,, Communications on Pure and Applied Analysis, 3 (2004), 607.  doi: 10.3934/cpaa.2004.3.607.  Google Scholar [6] P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations,, Advances in Differential Equations, 10 (2005), 153.   Google Scholar [7] P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators,, SIAM Journal on Control and Optimization, 47 (2008), 1.  doi: 10.1137/04062062X.  Google Scholar [8] L. C. Evans, Partial Differential Equations,, $1^{st}$ edition, (1998).   Google Scholar [9] P. Martinez, J. P. Raymond and J. Vancostenoble, Regional null controllability of a linearized Crocco-type equation,, SIAM Journal on Control and Optimization, 42 (2003), 709.  doi: 10.1137/S0363012902403547.  Google Scholar [10] P. Martinez, J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations,, Journal of Evolution Equations, 6 (2006), 325.  doi: 10.1007/s00028-006-0214-6.  Google Scholar [11] G. R. North, L. Howard, D. Pollard and B. Wielicki, Variational formulation of Budyko-Sellers climate models,, Journal of the Atmospheric Sciences, 36 (1979), 255.  doi: 10.1175/1520-0469(1979)036<0255:VFOBSC>2.0.CO;2.  Google Scholar [12] O. A. Ole\uinik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form,, American Mathematical Society, (1973).   Google Scholar [13] C. Wang, Approximate controllability of a class of degenerate systems,, Applied Mathematics and Computation, 203 (2008), 447.  doi: 10.1016/j.amc.2008.04.056.  Google Scholar [14] C. Wang, Approximate controllability of a class of semilinear systems with boundary degeneracy,, Journal of Evolution Equations, 10 (2010), 163.  doi: 10.1007/s00028-009-0044-4.  Google Scholar [15] J. Yin and C. Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy,, Journal of Differential Equations, 237 (2007), 421.  doi: 10.1016/j.jde.2007.03.012.  Google Scholar
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