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In this paper, we prove that if φ is a radially symmetric, sign-changing stationary solution of the nonlinear heat equationin the unit ball of R N , N ≥ 3, with Dirichlet boundary conditions, then the solution of (NLH) with initial value λφ blows up in finite time if |λ − 1| > 0 is sufficiently small and if α is subcritical and sufficiently close to 4/(N − 2).

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We prove that negative energy solutions of the complex Ginzburg-Landau equation e −iθ ut = ∆u + |u| α u blow up in finite time, where α > 0 and −π/2 < θ < π/2. For a fixed initial value u(0), we obtain estimates of the blow-up time T θ max as θ → ±π/2. It turns out that T θ max stays bounded (respectively, goes to infinity) as θ → ±π/2 in the case where the solution of the limiting nonlinear Schrödinger equation blows up in finite time (respectively, is global).2010 Mathematics Subject Classification. 35Q56, 35B44.