This paper is concerned with a supercritical semilinear diffusion equation. We show the existence of a solution that undergoes a birth-and-death process of a single peak emerging at arbitrarily prescribed positions and heights. In particular the solution has no asymptotic center of radial symmetry as time approaches infinity. We also construct a solution with arbitrarily prescribed grow-up set.
"Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation." Differential Integral Equations 17 (5-6) 535 - 548, 2004.