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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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