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January  2013, 12(1): 269-280. doi: 10.3934/cpaa.2013.12.269

Gradient blowup solutions of a semilinear parabolic equation with exponential source

1. 

College of Science, Xi’an Jiaotong University, Xi’an, 710049

2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

Received  June 2011 Revised  October 2011 Published  September 2012

In this paper, we consider the N-dimensional semilinear parabolic equation $ u_t=\Delta u+e^{|\nabla u|}$, for which the spatial derivative of solutions becomes unbounded in finite (or infinite) time while the solutions themselves remain bounded. We establish estimates of blowup rate as well as lower and upper bounds for the radial solutions. We prove that in this case the blowup rate does not match the one obtained by the rescaling method.
Citation: Zhengce Zhang, Yanyan Li. Gradient blowup solutions of a semilinear parabolic equation with exponential source. Communications on Pure and Applied Analysis, 2013, 12 (1) : 269-280. doi: 10.3934/cpaa.2013.12.269
References:
[1]

S. Angenent and M. Fila, Interior gradient blow-up in a semilinear parabolic equation, Differential Integral Equations, 9 (1996), 865-877.

[2]

J. S. Guo and B. Hu, Blowup rate for the heat equation in Lipschitz domains with nonlinear heat source terms on the boundary, J. Math. Anal. Appl., 269 (2002), 28-49. doi: 10.1016/S0022-247X(02)00002-1.

[3]

J. S. Guo and B. Hu, Blowup rate estimates for the heat equation with a nonlinear gradient source term, Discrete Contin. Dyn. Sys., 20 (2008), 927-937.

[4]

M. Fila and G. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations, 7 (1994), 811-821.

[5]

S. Filippas and R. V. Kohn, Refined asympotics for the blowup of $u_t-\Delta u=u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869. doi: 10.1002/cpa.3160450703.

[6]

A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.

[7]

Y. Giga and R. V. Kohn, Asympotically self-similar blow-up of semilinear heat equations, Commn. Pure Appl. Math., 38 (1985), 297-319. doi: 10.1002/cpa.3160380304.

[8]

M. A. Herrero and J. J. L. Velázquez, Blow-up profiles in one-dimensional semilinear parabolic problems, Commn. Partial Differential Equations, 17 (1992), 205-219. doi: 10.1080/03605309208820839.

[9]

M. A. Herrero and J. J. L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997.

[10]

M. A. Herrero and J. J. L. Velázquez, Genetic behaviour of one-dimensional blow-up patterns, Ann. Sc. Norm. Super Pisa CI. Sci., 19 (1992), 381-450.

[11]

M. A. Herrero and J. J. L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 10 (1993), 131-189.

[12]

M. Kardar, G. Parisi and Y. C. Zhang, Dynmic scailing of growing interfaces, Phys. Rev. Lett., 56 (1986), 889-892. doi: 10.1103/PhysRevLett.56.889.

[13]

J. Krug and H. Spohn, Universality classes for deterministic surface growth, Phys. Rev. A., 38 (1988), 4271-4283. doi: 10.1103/PhysRevA.38.4271.

[14]

O. A. Ladyženskaya and V. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc. Province, RI, 1967.

[15]

H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., 32 (1990), 262-288.

[16]

G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, Singapore, 1996. doi: 10.1142/3302.

[17]

A. Lunardi, "Analytic Semigroups and Optional Regularity in Parabolic Problems," Birkhauser, Basel, 1995.

[18]

A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, "Blow-up in Quasilinear Parabolic Equations" (Michael Grinfeld, Trans.), Walter de Gruyter, Berlin, 1995. doi: 10.1515/9783110889864.

[19]

Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, 15 (2002), 237-256.

[20]

Ph. Souplet and Q. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations, J. D'Analyse Math., 99 (2006), 335-396. doi: 10.1007/BF02789452.

[21]

Z. C. Zhang and B. Hu, Boundary gradient blowup in a semilinear parabolic equation, Discrete Contin. Dyn. Sys. A, 26 (2010), 767-779. doi: 10.3934/dcds.2010.26.767.

[22]

Z. C. Zhang and B. Hu, Rate estimate of gradient blowup for a heat equation with exponential Nonlinearity, Nonlinear Analysis, 72 (2010), 4594-4601. doi: 10.1016/j.na.2010.02.036.

show all references

References:
[1]

S. Angenent and M. Fila, Interior gradient blow-up in a semilinear parabolic equation, Differential Integral Equations, 9 (1996), 865-877.

[2]

J. S. Guo and B. Hu, Blowup rate for the heat equation in Lipschitz domains with nonlinear heat source terms on the boundary, J. Math. Anal. Appl., 269 (2002), 28-49. doi: 10.1016/S0022-247X(02)00002-1.

[3]

J. S. Guo and B. Hu, Blowup rate estimates for the heat equation with a nonlinear gradient source term, Discrete Contin. Dyn. Sys., 20 (2008), 927-937.

[4]

M. Fila and G. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations, 7 (1994), 811-821.

[5]

S. Filippas and R. V. Kohn, Refined asympotics for the blowup of $u_t-\Delta u=u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869. doi: 10.1002/cpa.3160450703.

[6]

A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.

[7]

Y. Giga and R. V. Kohn, Asympotically self-similar blow-up of semilinear heat equations, Commn. Pure Appl. Math., 38 (1985), 297-319. doi: 10.1002/cpa.3160380304.

[8]

M. A. Herrero and J. J. L. Velázquez, Blow-up profiles in one-dimensional semilinear parabolic problems, Commn. Partial Differential Equations, 17 (1992), 205-219. doi: 10.1080/03605309208820839.

[9]

M. A. Herrero and J. J. L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997.

[10]

M. A. Herrero and J. J. L. Velázquez, Genetic behaviour of one-dimensional blow-up patterns, Ann. Sc. Norm. Super Pisa CI. Sci., 19 (1992), 381-450.

[11]

M. A. Herrero and J. J. L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 10 (1993), 131-189.

[12]

M. Kardar, G. Parisi and Y. C. Zhang, Dynmic scailing of growing interfaces, Phys. Rev. Lett., 56 (1986), 889-892. doi: 10.1103/PhysRevLett.56.889.

[13]

J. Krug and H. Spohn, Universality classes for deterministic surface growth, Phys. Rev. A., 38 (1988), 4271-4283. doi: 10.1103/PhysRevA.38.4271.

[14]

O. A. Ladyženskaya and V. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc. Province, RI, 1967.

[15]

H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., 32 (1990), 262-288.

[16]

G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, Singapore, 1996. doi: 10.1142/3302.

[17]

A. Lunardi, "Analytic Semigroups and Optional Regularity in Parabolic Problems," Birkhauser, Basel, 1995.

[18]

A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, "Blow-up in Quasilinear Parabolic Equations" (Michael Grinfeld, Trans.), Walter de Gruyter, Berlin, 1995. doi: 10.1515/9783110889864.

[19]

Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, 15 (2002), 237-256.

[20]

Ph. Souplet and Q. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations, J. D'Analyse Math., 99 (2006), 335-396. doi: 10.1007/BF02789452.

[21]

Z. C. Zhang and B. Hu, Boundary gradient blowup in a semilinear parabolic equation, Discrete Contin. Dyn. Sys. A, 26 (2010), 767-779. doi: 10.3934/dcds.2010.26.767.

[22]

Z. C. Zhang and B. Hu, Rate estimate of gradient blowup for a heat equation with exponential Nonlinearity, Nonlinear Analysis, 72 (2010), 4594-4601. doi: 10.1016/j.na.2010.02.036.

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