Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Bounds for blow-up time for the heat equation under nonlinear boundary conditions

L. E. Paynea1 and P. W. Schaefera2

a1 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA, (

a2 Department of Mathematics, University of Tennessee, Knoxville, TN 37916, USA (


A differential inequality technique is used to determine a lower bound on the blow-up time for solutions to the heat equation subject to a nonlinear boundary condition when blow-up of the solution does occur. In addition, a sufficient condition which implies that blow-up does occur is determined.

(Received September 23 2008)

(Accepted January 22 2009)