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The Critical Exponent for Evolution Models with Power Non-linearity

  • Marcelo Rempel EbertEmail author
  • Linniker Monteiro Lourenço
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

In this note we derive Lr − Lq estimates for the solutions to the Cauchy problem
$$\displaystyle u_{tt} +(-\varDelta )^{\sigma } u = 0\,, \qquad t\geq 0, \ x\in {\mathbb {R}}^n, \qquad u(0,x)=0, \;\; u_t(0,x)=g(x), $$
with σ > 1. Moreover, we derived the critical index pc(n) for the existence of global in time small data solutions to the associated semilinear Cauchy problem with power nonlinearity |u|p, p > 1.

Notes

Acknowledgements

The first author has been has been partially supported by São Paulo Research Foundation (Fundação de Amparo à Pesquisa do Estado de São Paulo, FAPESP), grant 2017/19497-3. The second author was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Marcelo Rempel Ebert
    • 1
    Email author
  • Linniker Monteiro Lourenço
    • 2
  1. 1.Departamento de Computação e Matemática, FFCLRPUniversidade de São Paulo (USP)Ribeirão PretoBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de São CarlosSão CarlosBrazil

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