Green's function and infinite-time bubbling in the critical nonlinear heat equation

Carmen Cortazar, Monica Musso, Manuel Del Pino

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Abstract

Let Ω be a smooth bounded domain in ℝn, n ≥ 5. We consider the classical semilinear heat equation at the critical Sobolev exponent (Equation Presented) Let G(x, y) be the Dirichlet Green function of - A in Q and H(x, y) its regular part. Let qj e Q, j = 1,⋯, k, be points such that the matrix (Equation Presented) is positive definite. For any k ≥ 1 such points indeed exist. We prove the existence of a positive smooth solution u(x, t) which blows up by bubbling in infinite time near those points. More precisely, for large time t, u takes the approximate form (Equation Presented).

Original languageEnglish
Pages (from-to)283-344
Number of pages62
JournalJournal of the European Mathematical Society
Volume22
Issue number1
Early online date1 Oct 2019
DOIs
Publication statusPublished - 31 Jan 2020

Keywords

  • Critical exponent
  • Green's function
  • Infinite-time blow-up

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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