Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity

Abstract

We consider the semilinear heat equation with critical power nonlinearity. Using formal arguments based on matched asymptotic expansion techniques, we give a detailed description of radially symmetric sign‐changing solutions, which blow‐up at x = 0 and t = T < ∞, for space dimension N = 3,4,5,6. These solutions exhibit fast blow‐up; i.e. they satisfy limt↑T (T – t)1/(p‐1)u(0, t) = ∞. In contrast, radial solutions that are positive and decreasing behave as in the subcritical case for any N ⩾ 3. This last result is extended in the case of exponential nonlinearity and N = 2.