Rate estimates of gradient blowup for a heat equation with exponential nonlinearity. (English) Zbl 1189.35033

Summary: We consider a one-dimensional semilinear parabolic equation \(u_t = u_{xx}+\text e^{u_x}\), for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish estimates of blowup rate upper and lower bounds. We prove that in this case the blowup rate does not match the one obtained by the rescaling method.


35B44 Blow-up in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K58 Semilinear parabolic equations
Full Text: DOI


[1] Ladyz˘enskaya, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and Quasilinear Equations of Parabolic Type (1967), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[2] Lieberman, G. M., Second Order Parabolic Differential Equations (1996), World Scientific: World Scientific Singapore · Zbl 0884.35001
[3] Fila, M.; Lieberman, G., Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations, 7, 811-821 (1994) · Zbl 0811.35059
[4] Filippas, S.; Kohn, R. V., Refined asymptotics for the blowup of \(u_t - \Delta u = u^p\), Comm. Pure Appl. Math., 45, 821-869 (1992) · Zbl 0784.35010
[5] Friedman, A.; McLeod, J. B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-447 (1985) · Zbl 0576.35068
[6] Giga, Y.; Kohn, R. V., Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38, 297-319 (1985) · Zbl 0585.35051
[7] Herrero, M. A.; Velázquez, J. J.L., Blow-up profiles in one-dimensional semilinear parabolic problems, Comm. Partial Differential Equations, 17, 205-219 (1992) · Zbl 0772.35027
[8] Herrero, M. A.; Velázquez, J. J.L., Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5, 973-997 (1992) · Zbl 0767.35036
[9] Herrero, M. A.; Velázquez, J. J.L., Generic behaviour of one-dimensional blow-up patterns, Ann. Sc. Norm. Super Pisa Cl. Sci. (4), 19, 381-450 (1992) · Zbl 0798.35081
[10] Herrero, M. A.; Velázquez, J. J.L., Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 10, 131-189 (1993) · Zbl 0813.35007
[11] Levine, H. A., The role of critical exponents in blow-up theorems, SIAM Rev., 32, 262-288 (1990) · Zbl 0706.35008
[12] Samarskii, A. A.; Galaktionov, V. A.; Kurdyumov, S. P.; Mikhailov, A. P., Blow-up in Quasilinear Parabolic Equations (1995), Walter de Gruyter: Walter de Gruyter Berlin, (in Russian) · Zbl 1020.35001
[13] Alikakos, N.; Bates, P.; Grant, C., Blow up for a diffusion-advection equation, Proc. R. Soc. Edinburgh, A 113, 181-190 (1989) · Zbl 0707.35018
[14] Angenent, S.; Fila, M., Interior gradient blow-up in a semilinear parabolic equation, Differential Integral Equations, 9, 865-877 (1996) · Zbl 0864.35052
[15] Arrieta, J.; Rodriguez-Bernal, A.; Souplet, Ph., Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena, Ann. Sc. Norm. Super Pisa Cl. Sci. V. Sci. (5), 3, 1-15 (2004) · Zbl 1072.35098
[16] Asai, K.; Ishimura, N., On the interior derivative blow-up for the curvature evolution of capillary surfaces, Proc. Amer. Math. Soc., 126, 835-840 (1998) · Zbl 0992.35015
[17] Fila, M.; Taskinen, J.; Winkler, M., Convergence to a singular steady state of a parabolic equation with gradient blow-up, Appl. Math. Lett., 20, 578-582 (2007) · Zbl 1125.35367
[18] Giga, Y., Interior derivative blow-up for quasilinear parabolic equations, Discrete Contin. Dyn. Sys., 1, 449-461 (1995) · Zbl 0878.35015
[19] Souplet, Ph., Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, 15, 237-256 (2002) · Zbl 1015.35016
[20] Souplet, Ph.; Vazquez, J. L., Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem, Discrete Contin. Dyn. Sys. A, 14, 221-234 (2006) · Zbl 1116.35070
[21] Souplet, Ph.; Zhang, Q., Global solutions of inhomogeneous Hamilton-Jacobi equations, J. dÁnalyse Math., 99, 355-396 (2006) · Zbl 1149.35050
[22] Conner, G. R.; Grant, C. P., Asymptotics of blowup for a convection-diffusion equation with conservation, Differential Integral Equations, 9, 719-728 (1996) · Zbl 0856.35011
[23] Guo, J.-S.; Hu, B., Blowup rate estimates for the heat equation with a nonlinear gradient source term, Discrete Contin. Dyn. Sys., 20, 927-937 (2008) · Zbl 1159.35009
[24] Zhang, Z.-C.; Hu, B., Boundary gradient blowup in a semilinear parabolic equation, Discrete Contin. Dyn. Sys. A, 26, 767-779 (2010) · Zbl 1191.35074
[25] Guo, J.-S.; Hu, B., Blowup rate for heat equation in Lipschitz domains with nonlinear heat source terms on the boundary, J. Math. Anal. Appl., 269, 28-49 (2002) · Zbl 1006.35054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.