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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 & 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. Google Scholar [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. doi: 10.1016/S0022-247X(02)00002-1. Google Scholar [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. Google Scholar [4] M. Fila and G. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations,, Differential Integral Equations, 7 (1994), 811. Google Scholar [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. doi: 10.1002/cpa.3160450703. Google Scholar [6] A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425. Google Scholar [7] Y. Giga and R. V. Kohn, Asympotically self-similar blow-up of semilinear heat equations,, Commn. Pure Appl. Math., 38 (1985), 297. doi: 10.1002/cpa.3160380304. Google Scholar [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. doi: 10.1080/03605309208820839. Google Scholar [9] M. A. Herrero and J. J. L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations,, Differential Integral Equations, 5 (1992), 973. Google Scholar [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. Google Scholar [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. Google Scholar [12] M. Kardar, G. Parisi and Y. C. Zhang, Dynmic scailing of growing interfaces,, Phys. Rev. Lett., 56 (1986), 889. doi: 10.1103/PhysRevLett.56.889. Google Scholar [13] J. Krug and H. Spohn, Universality classes for deterministic surface growth,, Phys. Rev. A., 38 (1988), 4271. doi: 10.1103/PhysRevA.38.4271. Google Scholar [14] O. A. Ladyženskaya and V. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Amer. Math. Soc. Province, (1967). Google Scholar [15] H. A. Levine, The role of critical exponents in blow-up theorems,, SIAM Rev., 32 (1990), 262. Google Scholar [16] G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific, (1996). doi: 10.1142/3302. Google Scholar [17] A. Lunardi, "Analytic Semigroups and Optional Regularity in Parabolic Problems,", Birkhauser, (1995). Google Scholar [18] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, "Blow-up in Quasilinear Parabolic Equations", (Michael Grinfeld, (1995). doi: 10.1515/9783110889864. Google Scholar [19] Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions,, Differential Integral Equations, 15 (2002), 237. Google Scholar [20] Ph. Souplet and Q. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations,, J. D'Analyse Math., 99 (2006), 335. doi: 10.1007/BF02789452. Google Scholar [21] Z. C. Zhang and B. Hu, Boundary gradient blowup in a semilinear parabolic equation,, Discrete Contin. Dyn. Sys. A, 26 (2010), 767. doi: 10.3934/dcds.2010.26.767. Google Scholar [22] Z. C. Zhang and B. Hu, Rate estimate of gradient blowup for a heat equation with exponential Nonlinearity,, Nonlinear Analysis, 72 (2010), 4594. doi: 10.1016/j.na.2010.02.036. Google Scholar

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. Google Scholar [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. doi: 10.1016/S0022-247X(02)00002-1. Google Scholar [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. Google Scholar [4] M. Fila and G. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations,, Differential Integral Equations, 7 (1994), 811. Google Scholar [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. doi: 10.1002/cpa.3160450703. Google Scholar [6] A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425. Google Scholar [7] Y. Giga and R. V. Kohn, Asympotically self-similar blow-up of semilinear heat equations,, Commn. Pure Appl. Math., 38 (1985), 297. doi: 10.1002/cpa.3160380304. Google Scholar [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. doi: 10.1080/03605309208820839. Google Scholar [9] M. A. Herrero and J. J. L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations,, Differential Integral Equations, 5 (1992), 973. Google Scholar [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. Google Scholar [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. Google Scholar [12] M. Kardar, G. Parisi and Y. C. Zhang, Dynmic scailing of growing interfaces,, Phys. Rev. Lett., 56 (1986), 889. doi: 10.1103/PhysRevLett.56.889. Google Scholar [13] J. Krug and H. Spohn, Universality classes for deterministic surface growth,, Phys. Rev. A., 38 (1988), 4271. doi: 10.1103/PhysRevA.38.4271. Google Scholar [14] O. A. Ladyženskaya and V. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Amer. Math. Soc. Province, (1967). Google Scholar [15] H. A. Levine, The role of critical exponents in blow-up theorems,, SIAM Rev., 32 (1990), 262. Google Scholar [16] G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific, (1996). doi: 10.1142/3302. Google Scholar [17] A. Lunardi, "Analytic Semigroups and Optional Regularity in Parabolic Problems,", Birkhauser, (1995). Google Scholar [18] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, "Blow-up in Quasilinear Parabolic Equations", (Michael Grinfeld, (1995). doi: 10.1515/9783110889864. Google Scholar [19] Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions,, Differential Integral Equations, 15 (2002), 237. Google Scholar [20] Ph. Souplet and Q. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations,, J. D'Analyse Math., 99 (2006), 335. doi: 10.1007/BF02789452. Google Scholar [21] Z. C. Zhang and B. Hu, Boundary gradient blowup in a semilinear parabolic equation,, Discrete Contin. Dyn. Sys. A, 26 (2010), 767. doi: 10.3934/dcds.2010.26.767. Google Scholar [22] Z. C. Zhang and B. Hu, Rate estimate of gradient blowup for a heat equation with exponential Nonlinearity,, Nonlinear Analysis, 72 (2010), 4594. doi: 10.1016/j.na.2010.02.036. Google Scholar
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