Stability and asymptotic behaviour of solutions of the heat equation

where Ω is a bounded and smooth subset of Rn , n 1, m > 0 and p 1. Problem (1.1)–(1.3) (see Galaktionov, 1981; Samarskii et al., 1995) describes the propagation of thermal perturbations in a medium with a nonlinear heat conduction coefficient and a heat source depending on the temperature when u0 0. Local existence for the solutions of (1.1)–(1.3) has been proved when m > 1 (the so-called slow diffusion case) in Galaktionov (1981), Levine & Saks (1984), Nakao (1983), Samarskii et al. (1995) and when 0 < m < 1 (the fast diffusion case) in Filo (1987). The same type of results holds for the heat equation with source, when m = 1. See for example Ball (1977), Fujita (1966, 1968), Levine (1973), Tsutsumi (1972). However, other results are known for the heat equation when 1 < p n+2 n−2 (the last condition being necessary only when n 3) and u0 ∈ H1 0 (Ω). For large initial data u0 in some sense, it is well known that the solution u of (1.1)–(1.3) with m = 1 blows up in a finite time (see Ikehata & Suzuki, 2000), meanwhile for small initial data, exponentially decaying solutions are obtained (see Ikehata & Suzuki, 2000 and the references therein). In a recent paper, Ikehata (2000) showed that all the global solutions for (1.1)–(1.3) with m = 1 naturally contain a Palais–Smale sequence so that the global compactness result due to Struwe (1984) can be applied to this functional sequence (see also Cerami et al., 1986). In Section 2 we consider the non-dimensionalized heat equation with boundary