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In this note we obtain estimates in terms of the size of the initial data for the blow-up time of positive solutions of the heat equation in R+ with a nonlinear boundary condition −ux(0, t) = up(0, t). Introduction. In this note we obtain estimates for the blow-up time of positive solutions of the following parabolic problem (1)  ut = uxx in R+ × (0, Tλ), −ux(0, t) = u(0, t) in (0, Tλ), u(x, 0) = λφ(x) > 0 in R+. where p > 1 is fixed and λ > 0 is a parameter. Throughout this note we assume that the initial datum φ is continuous, positive and bounded. Existence, uniqueness, regularity and continuous dependence on the initial data for this problem were proved, for instance, in [2]. For problem (1), it is well known that if λ is large enough the solution blows up in finite time Tλ (Tλ depends on λ) , if and only if p > 1, see for example, [1], [3], [4], [8], [10]. This means that there exists a finite time Tλ with lim t↗Tλ ‖u(·, t)‖∞ = +∞. Here we are interested in the asymptotic behaviour of Tλ when λ goes to infinity. We prove the following Theorem, Theorem 1. Under the above assumptions on φ, the function λ 7→ Tλ is decreasing and continuous with the following asymptotic behaviour at infinity, lim λ→∞ λTλ = T0. Here T0 is the blow-up time of the solution of (1) with initial datum u(x, 0) ≡ φ(0). Some related papers that deal with the heat equation with a nonlinear source in the entire space are [7] and [9]. 1991 Mathematics Subject Classification. 35B40, 35K60, 35B30.

Life span for solutions of the heat equation with a nonlinear boundary condition