Blow‐up profile for radial solutions of the nonlinear heat equation

Article type: Research Article

Authors: Ramiandrisoa, Arthur

Affiliations: Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris Cedex 05, France

Abstract: Assuming that the maximal classical solution u of u_t-\Delta u=u^p on B(0,R) has, at time T_{\max}, a blow‐up set \mathbf{B}(u) that is not the singleton \left\{0\right\}, we prove that {\|u\|}_{L^q (\varOmega)} blows up for all q>(p-1)/2, the energy blows up and u blows up completely after T_{\max} in the sense of Baras–Cohen (equivalent to the fact that there is no weak solution extending u beyond T_{\max}). We also study this type of blow‐up through numerical experiments as well as degenerate blow‐up.

Keywords: Blow‐up, subcritical norms, radial solutions, heat equation, blow‐up set, numerical experiments

Journal: Asymptotic Analysis, vol. 21, no. 3‐4, pp. 221-238, 1999