Asymptotic behavior of solutions to a class of semilinear parabolic equations with boundary degeneracy

Wang, Chunpeng

doi:10.1090/S0002-9939-2013-11945-3

Asymptotic behavior of solutions to a class of semilinear parabolic equations with boundary degeneracy
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Proc. Amer. Math. Soc. 141 (2013), 3125-3140 Request permission

Abstract:

This paper concerns the asymptotic behavior of solutions to a semilinear parabolic equation with boundary degeneracy. It is proved that for the problem in a bounded domain with a homogeneous boundary condition, there exist both nontrivial global and blowing-up solutions if the degeneracy is not strong, while the nontrivial solution must blow up in a finite time if the degeneracy is strong enough. For the problem in an unbounded domain, blowing-up theorems of Fujita type are established and the critical Fujita exponent is finite in the not strong degeneracy case, while infinite in the other case. Furthermore, the behavior of solutions at the degenerate point is studied, and it is shown that for the nontrivial initial datum vanishing at the degenerate point, the solution always vanishes at the degenerate point if the degeneracy is strong enough, while never if it is not.

References

F. Alabau-Boussouira, P. Cannarsa, and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ. 6 (2006), no. 2, 161–204. MR 2227693, DOI 10.1007/s00028-006-0222-6
Black, F. and Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy, 81(1973), 637–659.
P. Cannarsa, G. Fragnelli, and J. Vancostenoble, Linear degenerate parabolic equations in bounded domains: controllability and observability, Systems, control, modeling and optimization, IFIP Int. Fed. Inf. Process., vol. 202, Springer, New York, 2006, pp. 163–173. MR 2241704, DOI 10.1007/0-387-33882-9_{1}5
P. Cannarsa, G. Fragnelli, and J. Vancostenoble, Regional controllability of semilinear degenerate parabolic equations in bounded domains, J. Math. Anal. Appl. 320 (2006), no. 2, 804–818. MR 2225996, DOI 10.1016/j.jmaa.2005.07.006
Piermarco Cannarsa, Patrick Martinez, and Judith Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations, Commun. Pure Appl. Anal. 3 (2004), no. 4, 607–635. MR 2106292, DOI 10.3934/cpaa.2004.3.607
P. Cannarsa, P. Martinez, and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations 10 (2005), no. 2, 153–190. MR 2106129
P. Cannarsa, P. Martinez, and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim. 47 (2008), no. 1, 1–19. MR 2373460, DOI 10.1137/04062062X
Keng Deng and Howard A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000), no. 1, 85–126. MR 1742850, DOI 10.1006/jmaa.1999.6663
Hiroshi Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t}=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124 (1966). MR 214914
Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), no. 2, 262–288. MR 1056055, DOI 10.1137/1032046
P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ. 6 (2006), no. 2, 325–362. MR 2227700, DOI 10.1007/s00028-006-0214-6
North, G. R., Howard, L., Pollard, D. and Wielicki, B., Variational formulation of Budyko-Sellers climate models, J. Atmos. Sci., 36(1979), 255–259.
Chunpeng Wang, Approximate controllability of a class of semilinear systems with boundary degeneracy, J. Evol. Equ. 10 (2010), no. 1, 163–193. MR 2602931, DOI 10.1007/s00028-009-0044-4
Yan Feng Wei and Run Mei Du, Null controllability of a class of parabolic systems with boundary degeneracy, J. Jilin Univ. Sci. 49 (2011), no. 5, 849–852 (Chinese, with English and Chinese summaries). MR 2885035
Zhuoqun Wu, Junning Zhao, Jingxue Yin, and Huilai Li, Nonlinear diffusion equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. Translated from the 1996 Chinese original and revised by the authors. MR 1881297, DOI 10.1142/9789812799791
Jingxue Yin and Chunpeng Wang, Properties of the boundary flux of a singular diffusion process, Chinese Ann. Math. Ser. B 25 (2004), no. 2, 175–182. MR 2055268, DOI 10.1142/S0252959904000184
Jingxue Yin and Chunpeng Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy, J. Differential Equations 237 (2007), no. 2, 421–445. MR 2330953, DOI 10.1016/j.jde.2007.03.012