This paper is concerned with positive solutions of the semilinear diffusion equation in under the Neumann boundary condition, where is a constant and is a bounded domain in with boundary. This equation has the constant solution with the blow-up time . It is shown that for any and open cone in , there exists a positive function in with on and such that the blow-up time of the solution with initial data is larger than and the function belongs to the cone . A theorem on the blow-up profile is also given.
Hiroki YAGISITA. "Variable instability of a constant blow-up solution in a nonlinear heat equation." J. Math. Soc. Japan 56 (4) 1007 - 1017, October, 2004. https://doi.org/10.2969/jmsj/1190905446