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Published in:
On Effective PDEs of Quantum Physics Read chapter Read first chapter

2019 | OriginalPaper | Chapter

The Critical Exponent for Evolution Models with Power Non-linearity

Authors : Marcelo Rempel Ebert, Linniker Monteiro Lourenço

Published in: New Tools for Nonlinear PDEs and Application

Publisher: Springer International Publishing

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Abstract

In this note we derive L r − L q 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 p c(n) for the existence of global in time small data solutions to the associated semilinear Cauchy problem with power nonlinearity |u|p, p > 1.

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previous chapter Incompressible Limits for Generalisations to Symmetrisable Systems
next chapter Blow-Up or Global Existence for the Fractional Ginzburg-Landau Equation in Multi-dimensional Case
Appendix
Available only for authorised users
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Metadata
Title
The Critical Exponent for Evolution Models with Power Non-linearity
Authors
Marcelo Rempel Ebert
Linniker Monteiro Lourenço
Copyright Year
2019
Book
New Tools for Nonlinear PDEs and Application

Print ISBN: 978-3-030-10936-3

Electronic ISBN: 978-3-030-10937-0

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-10937-0_5

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