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Journal of Elliptic and Parabolic Equations

, Volume 4, Issue 1, pp 141–176 | Cite as

The nonlinear heat equation involving highly singular initial values and new blowup and life span results

  • Slim Tayachi
  • Fred B. Weissler
Article
  • 46 Downloads

Abstract

In this paper we prove local existence of solutions to the nonlinear heat equation \(u_t = \Delta u +a |u|^\alpha u, \; t\in (0,T),\; x=(x_1,\ldots , x_N)\in {\mathbb {R}}^N,\; a = \pm 1,\; \alpha >0;\) with initial value \(u(0)\in L^1_{\mathrm{{loc}}}({\mathbb {R}}^N{\setminus }\{0\}),\) anti-symmetric with respect to \(x_1,\; x_2,\ldots , x_m\) and \(|u(0)|\le C(-1)^m\partial _{1}\partial _{2}\cdots \partial _{m}(|x|^{-\gamma })\) for \(x_1>0,\ldots , x_m>0,\) where \(C>0\) is a constant, \(m\in \{1, 2,\ldots , N\},\) \(0<\gamma <N\) and \(0<\alpha <2/(\gamma +m).\) This gives a local existence result with highly singular initial values. As an application, for \(a=1,\) we establish new blowup criteria for \(0<\alpha \le 2/(\gamma +m),\) including the case \(m=0.\) Moreover, if \((N-4)\alpha <2,\) we prove the existence of initial values \(u_0 = \lambda f,\) for which the resulting solution blows up in finite time \(T_{\max }(\lambda f),\) if \(\lambda >0\) is sufficiently small. We also construct blowing up solutions with initial data \(\lambda _n f\) such that \(\lambda _n^{[({1\over \alpha }-{\gamma +m\over 2})^{-1}]}T_{\max }(\lambda _n f)\) has different finite limits along different sequences \(\lambda _n\rightarrow 0.\) Our result extends the known “small lambda” blow up results for new values of \(\alpha\) and a new class of initial data.

Keywords

Nonlinear heat equation Highly singular initial values Finite time blow-up 

Mathematics Subject Classification

Primary 35K55 35A01 35B44 Secondary 35K57 35C15 

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Copyright information

© Orthogonal Publishing and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire Équations aux Dérivées Partielles LR03ES04, Département de Mathématiques, Faculté des Sciences de TunisUniversité de Tunis El ManarTunisTunisia
  2. 2.Université Paris 13, Sorbonne Paris Cité, CNRS UMR 7539 LAGAVilletaneuseFrance

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