Boundedness of global solutions for a heat equation with exponential gradient source. (English) Zbl 1254.35122

Summary: We consider a one-dimensional semilinear parabolic equation with exponential gradient source and provide a complete classification of large time behavior of the classical solutions: either the space derivative of the solution blows up in finite time with the solution itself remaining bounded, or the solution is global and converges in \(C^1\) norm to the unique steady state. The main difficulty is to prove \(C^1\) boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov’s functional by carrying out the method of Zelenyak.


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


[1] P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, vol. 62 of Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1982.
[2] M. Kardar, G. Parisi, and Y.-C. Zhang, “Dynamic scaling of growing interfaces,” Physical Review Letters, vol. 56, no. 9, pp. 889-892, 1986. · Zbl 1101.82329
[3] T. I. Zelenyak, “Stabilisation of solutions of boundary value problems for a second-order equation with one space variable,” Differential Equations, vol. 4, pp. 17-22, 1968. · Zbl 0232.35053
[4] Z. Zhang and B. Hu, “Rate estimates of gradient blowup for a heat equation with exponential nonlinearity,” Nonlinear Analysis, vol. 72, no. 12, pp. 4594-4601, 2010. · Zbl 1189.35033
[5] J. M. Arrieta, A. R. Bernal, and P. Souplet, “Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, vol. 5, pp. 1-15, 2004. · Zbl 1072.35098
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.