Blow-up for the heat equation with a general memory boundary condition

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September 2012, 11(5): 2147-2156. doi: 10.3934/cpaa.2012.11.2147

Blow-up for the heat equation with a general memory boundary condition

1.	Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
2.	Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, United States

Received March 2011 Revised November 2011 Published March 2012

Abstract
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In this paper, we study the long-time behavior of nonnegative solutions of the heat equation with a general memory boundary condition. We first present conditions on the memory term for finite time blow-up. We then establish global existence results through both analytical and numerical methods. Finally, we show that under certain conditions blow-up occurs only on the boundary.

Keywords: finite time blow-up, General memory boundary condition, blow-up on boundary., global existence.

Mathematics Subject Classification: Primary: 35B44, 35K05, 35K2.

Citation: Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147

References:

[1]	J. R. Anderson, K. Deng and Z. Dong, Global solvability for the heat equation with boundary flux governed by nonlinear memory,, Quart. Appl. Math., 69 (2011), 759. doi: 10.1090/S0033-569X-2011-01238-X.
[2]	M. Ciarletta, A differential problem for heat equation with a boundary condition with memory,, Appl. Math. Letters, 10 (1997), 95. doi: 10.1016/S0893-9659(96)00118-8.
[3]	K. Deng and M. Xu, On solutions of a singular diffusion equation,, Nonlinear Anal., 41 (2000), 489. doi: 10.1016/S0362-546X(98)00292-2.
[4]	M. Fabrizio and A. Morro, A boundary condition with memory in electromagnetism,, Arch. Rational Mech. Anal., 136 (1996), 359. doi: 10.1007/BF02206624.
[5]	R. Ferreira, P. Groisman and J. D. Rossi, Numerical blow-up for a nonlinear problem with a nonlinear boundary condition,, Math. Models Methods Appl. Sci., 12 (2002), 461. doi: 10.1142/S021820250200174X.
[6]	B. Hu and H.-M. Yin, The profile near blowup time for solutions of the heat equation with a nonlinear boundary condition,, Trans. Amer. Math. Soc., 346 (1994), 117. doi: 10.2307/2154944.
[7]	B. Hu and H.-M. Yin, Critical exponents for a system of heat equations coupled in a non-linear boundary condition,, Math. Methods Appl. Sci., 19 (1996), 1099. doi: 10.1002/(SICI)1099-1476(19960925)19:14<1099::AID-MMA780>3.0.CO;2-J.
[8]	H. A. Levine, S. Pamuk, B. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma,, Bull. Math. Biol., 63 (2001), 801. doi: 10.1006/bulm.2001.0240.
[9]	H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time,, J. Differential Equations, 16 (1974), 319. doi: 10.1016/0022-0396(74)90018-7.
[10]	J. López-Gómez, V. Márquez and N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition,, J. Differential Equations, 92 (1991), 384.
[11]	L. E. Payne and P. W. Schaefer, Bounds for blow-up time for the heat equation under nonlinear boundary conditions,, Proc. Royal Soc. Edinburgh Sect. A, 139 (2009), 1289. doi: 10.1017/S0308210508000802.
[12]	D. F. Rial and J. D. Rossi, Blow-up results and localization of blow-up points in an N-dimensional smooth domain,, Duke Math. J., 88 (1997), 391. doi: 10.1215/S0012-7094-97-08816-5.
[13]	W. Walter, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition,, SIAM J. Math. Anal., 6 (1975), 85. doi: 10.1137/0506008.
[14]	M. X. Wang and Y. H. Wu, Global existence and blow up problems for quasilinear parabolic equations with nonlinear boundary conditions,, SIAM J. Math. Anal., 24 (1993), 1515. doi: 10.1137/0524085.

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References:

[1]	J. R. Anderson, K. Deng and Z. Dong, Global solvability for the heat equation with boundary flux governed by nonlinear memory,, Quart. Appl. Math., 69 (2011), 759. doi: 10.1090/S0033-569X-2011-01238-X.
[2]	M. Ciarletta, A differential problem for heat equation with a boundary condition with memory,, Appl. Math. Letters, 10 (1997), 95. doi: 10.1016/S0893-9659(96)00118-8.
[3]	K. Deng and M. Xu, On solutions of a singular diffusion equation,, Nonlinear Anal., 41 (2000), 489. doi: 10.1016/S0362-546X(98)00292-2.
[4]	M. Fabrizio and A. Morro, A boundary condition with memory in electromagnetism,, Arch. Rational Mech. Anal., 136 (1996), 359. doi: 10.1007/BF02206624.
[5]	R. Ferreira, P. Groisman and J. D. Rossi, Numerical blow-up for a nonlinear problem with a nonlinear boundary condition,, Math. Models Methods Appl. Sci., 12 (2002), 461. doi: 10.1142/S021820250200174X.
[6]	B. Hu and H.-M. Yin, The profile near blowup time for solutions of the heat equation with a nonlinear boundary condition,, Trans. Amer. Math. Soc., 346 (1994), 117. doi: 10.2307/2154944.
[7]	B. Hu and H.-M. Yin, Critical exponents for a system of heat equations coupled in a non-linear boundary condition,, Math. Methods Appl. Sci., 19 (1996), 1099. doi: 10.1002/(SICI)1099-1476(19960925)19:14<1099::AID-MMA780>3.0.CO;2-J.
[8]	H. A. Levine, S. Pamuk, B. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma,, Bull. Math. Biol., 63 (2001), 801. doi: 10.1006/bulm.2001.0240.
[9]	H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time,, J. Differential Equations, 16 (1974), 319. doi: 10.1016/0022-0396(74)90018-7.
[10]	J. López-Gómez, V. Márquez and N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition,, J. Differential Equations, 92 (1991), 384.
[11]	L. E. Payne and P. W. Schaefer, Bounds for blow-up time for the heat equation under nonlinear boundary conditions,, Proc. Royal Soc. Edinburgh Sect. A, 139 (2009), 1289. doi: 10.1017/S0308210508000802.
[12]	D. F. Rial and J. D. Rossi, Blow-up results and localization of blow-up points in an N-dimensional smooth domain,, Duke Math. J., 88 (1997), 391. doi: 10.1215/S0012-7094-97-08816-5.
[13]	W. Walter, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition,, SIAM J. Math. Anal., 6 (1975), 85. doi: 10.1137/0506008.
[14]	M. X. Wang and Y. H. Wu, Global existence and blow up problems for quasilinear parabolic equations with nonlinear boundary conditions,, SIAM J. Math. Anal., 24 (1993), 1515. doi: 10.1137/0524085.

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