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

2019 | OriginalPaper | Chapter

Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

Author : Alessandro Palmieri

Published in: New Tools for Nonlinear PDEs and Application

Publisher: Springer International Publishing

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Abstract

We consider a semilinear wave equation with scale-invariant damping and mass and power nonlinearity. For this model we prove some global (in time) existence results in odd spatial dimension n, under the assumption that the multiplicative constants μ and ν 2, which appear in the coefficients of the damping and of the mass terms, respectively, satisfy an interplay condition which makes the model somehow “wave-like”. Combining these global existence results with a recently proved blow-up result, we will find as critical exponent for the considered model the largest between suitable shifts of the Strauss exponent and of Fujita exponent, respectively. Besides, the competition among these two kind of exponents shows how the interrelationship between μ and ν 2 determines the possible transition from a “hyperbolic-like” to a “parabolic-like” model. Nevertheless, in the case n ≥ 3 we will restrict our considerations to the radial symmetric case.

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previous chapter The Cauchy Problem for Dissipative Wave Equations with Weighted Nonlinear Terms
next chapter Wave Equations in Modulation Spaces–Decay Versus Loss of Regularity
Literature
1.
F. Asakura, Existence of a global solution to a semilinear wave equation with slowly decreasing initial data in three space dimensions. Commun. Partial Differ. Equ. 11(13), 1459–1487 (1986)MathSciNetCrossRef
2.
R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. 2. (Interscience, New York, 1962)MATH
3.
M. D’Abbicco, The threshold of effective damping for semilinear wave equation. Math. Methods Appl. Sci. 38(6), 1032–1045 (2015)MathSciNetCrossRef
4.
M. D’Abbicco, S. Lucente, A modified test function method for damped wave equation. Adv. Nonlinear Stud. 13(4), 867–892 (2013)MathSciNetCrossRef
5.
M. D’Abbicco, S. Lucente, NLWE with a special scale invariant damping in odd space dimension. Discret. Contin. Dyn. Syst. Suppl. (2015). 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, pp. 312–319
6.
M. D’Abbicco, S. Lucente, M. Reissig, A shift in the Strauss exponent for semi-linear wave equations with a not effective damping. J. Differ. Equ. 259(10), 5040–5073 (2015)CrossRef
7.
M. D’Abbicco, A. Palmieri, L p − L q estimates on the conjugate line for semilinear critical dissipative Klein-Gordon equations. Preprint
8.
9.
M. Kato, M. Sakuraba, Global existence and blow-up for semilinear damped wave equations in three space dimensions (2018). arXiv:1807.04327
10.
H. Kubo, Asymptotic behaviour of solutions to semilinear wave equations with initial data of slow decay. Math. Methods Appl. Sci. 17(12), 953–970 (1994)MathSciNetCrossRef
11.
12.
H. Kubo, Slowly decaying solutions for semilinear wave equations in odd space dimensions. Nonlinear Anal. 28(2), 327–357 (1997)MathSciNetCrossRef
13.
H. Kubo, K. Kubota, Asymptotic behaviors of radially symmetric solutions of \(\square u = |u|{ }^p\) for super critical values p in odd space dimensions. Hokkaido Math. J. 24(2), 287–336 (1995)
14.
H. Kubo, K. Kubota, Asymptotic behaviors of radial solutions to semilinear wave equations in odd space dimensions. Hokkaido Math. J. 24(1), 9–51 (1995)MathSciNetCrossRef
15.
N.A. Lai, Weighted L 2 − L 2 estimate for wave equation and its applications (2018). arXiv:1807.05109
16.
N.A. Lai, H. Takamura, K. Wakasa, Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent. J. Differ. Equ. 263(9), 5377–5394 (2017)MathSciNetCrossRef
17.
W. Nunes do Nascimento, A. Palmieri, M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation. Math. Nachr. 290(11–12), 1779–1805 (2017)
18.
A. Palmieri, Global existence of solutions for semi-linear wave equation with scale-invariant damping and mass in exponentially weighted spaces. J. Math. Anal. Appl. 461(2), 1215–1240 (2018)MathSciNetCrossRef
19.
A. Palmieri, M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, II. Math. Nachr. 291(11–12), 1859–1892 (2018)MathSciNetCrossRef
20.
21.
A. Palmieri, A global existence result for a semilinear wave equation with scale-invariant damping and mass in even space dimension (2018). arxiv:1804.03978
22.
A. Palmieri, Global in time existence and blow-up results for a semilinear wave equation with scale-invariant damping and mass. PhD thesis, TU Bergakademie Freiberg, 2018, 279pp
23.
24.
H. Takamura, Improved Kato’s lemma on ordinary differential inequality and its application to semilinear wave equations. Nonlinear Anal. 125, 227–240 (2015)MathSciNetCrossRef
25.
Z. Tu, J. Lin, A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent (2017). arxiv:1709.00866v2
26.
Z. Tu, J. Lin, Life-span of semilinear wave equations with scale-invariant damping: critical strauss exponent case (2017). arxiv:1711.00223
27.
Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping, in Fourier Analysis. Trends Mathematics (Birkhäuser/Springer, Cham, 2014), pp. 375–390
Metadata
Title
Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension
Author
Alessandro Palmieri
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_12

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