# 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 and Continuous Dynamical Systems, 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), 161-204. doi: 10.1007/s00028-006-0222-6. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [3] P. Cannarsa, G. Fragnelli and J. Vancostenoble, Linear degenerate parabolic equations in bounded domains: Controllability and observability, in Systems, control, modeling and optimization (eds. F. Ceragioli, A. Dontchev, H. Furuta, K. Marti and L. Pandolfi), IFIP International Federation for Information Processing, Springer, New York, 202 (2006), 163-173. doi: 10.1007/0-387-33882-9_15. [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), 804-818. doi: 10.1016/j.jmaa.2005.07.006. [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), 607-635. doi: 10.3934/cpaa.2004.3.607. [6] P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Advances in Differential Equations, 10 (2005), 153-190. [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), 1-19. doi: 10.1137/04062062X. [8] L. C. Evans, Partial Differential Equations, $1^{st}$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. [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), 709-728. doi: 10.1137/S0363012902403547. [10] P. Martinez, J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, Journal of Evolution Equations, 6 (2006), 325-362. doi: 10.1007/s00028-006-0214-6. [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), 255-259. doi: 10.1175/1520-0469(1979)036<0255:VFOBSC>2.0.CO;2. [12] O. A. Ole\uinik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, American Mathematical Society, Rhode Island and Plenum Press, New York, 1973. [13] C. Wang, Approximate controllability of a class of degenerate systems, Applied Mathematics and Computation, 203 (2008), 447-456. doi: 10.1016/j.amc.2008.04.056. [14] C. Wang, Approximate controllability of a class of semilinear systems with boundary degeneracy, Journal of Evolution Equations, 10 (2010), 163-193. doi: 10.1007/s00028-009-0044-4. [15] J. Yin and C. Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy, Journal of Differential Equations, 237 (2007), 421-445. doi: 10.1016/j.jde.2007.03.012.

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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), 161-204. doi: 10.1007/s00028-006-0222-6. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [3] P. Cannarsa, G. Fragnelli and J. Vancostenoble, Linear degenerate parabolic equations in bounded domains: Controllability and observability, in Systems, control, modeling and optimization (eds. F. Ceragioli, A. Dontchev, H. Furuta, K. Marti and L. Pandolfi), IFIP International Federation for Information Processing, Springer, New York, 202 (2006), 163-173. doi: 10.1007/0-387-33882-9_15. [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), 804-818. doi: 10.1016/j.jmaa.2005.07.006. [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), 607-635. doi: 10.3934/cpaa.2004.3.607. [6] P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Advances in Differential Equations, 10 (2005), 153-190. [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), 1-19. doi: 10.1137/04062062X. [8] L. C. Evans, Partial Differential Equations, $1^{st}$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. [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), 709-728. doi: 10.1137/S0363012902403547. [10] P. Martinez, J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, Journal of Evolution Equations, 6 (2006), 325-362. doi: 10.1007/s00028-006-0214-6. [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), 255-259. doi: 10.1175/1520-0469(1979)036<0255:VFOBSC>2.0.CO;2. [12] O. A. Ole\uinik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, American Mathematical Society, Rhode Island and Plenum Press, New York, 1973. [13] C. Wang, Approximate controllability of a class of degenerate systems, Applied Mathematics and Computation, 203 (2008), 447-456. doi: 10.1016/j.amc.2008.04.056. [14] C. Wang, Approximate controllability of a class of semilinear systems with boundary degeneracy, Journal of Evolution Equations, 10 (2010), 163-193. doi: 10.1007/s00028-009-0044-4. [15] J. Yin and C. Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy, Journal of Differential Equations, 237 (2007), 421-445. doi: 10.1016/j.jde.2007.03.012.
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