American Institute of Mathematical Sciences

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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

Keng Deng 1, and  Zhihua Dong 2,

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

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), 759-770. doi: 10.1090/S0033-569X-2011-01238-X.  Google Scholar

[2]

M. Ciarletta, A differential problem for heat equation with a boundary condition with memory, Appl. Math. Letters, 10 (1997), 95-101. doi: 10.1016/S0893-9659(96)00118-8.  Google Scholar

[3]

K. Deng and M. Xu, On solutions of a singular diffusion equation, Nonlinear Anal., 41 (2000), 489-500. doi: 10.1016/S0362-546X(98)00292-2.  Google Scholar

[4]

M. Fabrizio and A. Morro, A boundary condition with memory in electromagnetism, Arch. Rational Mech. Anal., 136 (1996), 359-381. doi: 10.1007/BF02206624.  Google Scholar

[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), 461-483. doi: 10.1142/S021820250200174X.  Google Scholar

[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), 117-135. doi: 10.2307/2154944.  Google Scholar

[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), 1099-1120. doi: 10.1002/(SICI)1099-1476(19960925)19:14<1099::AID-MMA780>3.0.CO;2-J.  Google Scholar

[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), 801-863. doi: 10.1006/bulm.2001.0240.  Google Scholar

[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), 319-334. doi: 10.1016/0022-0396(74)90018-7.  Google Scholar

[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), 384-401.  Google Scholar

[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), 1289-1296. doi: 10.1017/S0308210508000802.  Google Scholar

[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), 391-405. doi: 10.1215/S0012-7094-97-08816-5.  Google Scholar

[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), 85-90. doi: 10.1137/0506008.  Google Scholar

[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), 1515-1521. doi: 10.1137/0524085.  Google Scholar

show all references

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), 759-770. doi: 10.1090/S0033-569X-2011-01238-X.  Google Scholar

[2]

M. Ciarletta, A differential problem for heat equation with a boundary condition with memory, Appl. Math. Letters, 10 (1997), 95-101. doi: 10.1016/S0893-9659(96)00118-8.  Google Scholar

[3]

K. Deng and M. Xu, On solutions of a singular diffusion equation, Nonlinear Anal., 41 (2000), 489-500. doi: 10.1016/S0362-546X(98)00292-2.  Google Scholar

[4]

M. Fabrizio and A. Morro, A boundary condition with memory in electromagnetism, Arch. Rational Mech. Anal., 136 (1996), 359-381. doi: 10.1007/BF02206624.  Google Scholar

[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), 461-483. doi: 10.1142/S021820250200174X.  Google Scholar

[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), 117-135. doi: 10.2307/2154944.  Google Scholar

[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), 1099-1120. doi: 10.1002/(SICI)1099-1476(19960925)19:14<1099::AID-MMA780>3.0.CO;2-J.  Google Scholar

[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), 801-863. doi: 10.1006/bulm.2001.0240.  Google Scholar

[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), 319-334. doi: 10.1016/0022-0396(74)90018-7.  Google Scholar

[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), 384-401.  Google Scholar

[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), 1289-1296. doi: 10.1017/S0308210508000802.  Google Scholar

[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), 391-405. doi: 10.1215/S0012-7094-97-08816-5.  Google Scholar

[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), 85-90. doi: 10.1137/0506008.  Google Scholar

[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), 1515-1521. doi: 10.1137/0524085.  Google Scholar

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