Critical regularity of nonlinearities in semilinear effectively damped wave models

Abdelhamid Mohammed Djaouti; Michael Reissig; Abdelhamid Mohammed Djaouti; Michael Reissig

doi:10.3934/math.2023236

AIMS Mathematics

2023, Volume 8, Issue 2: 4764-4785. doi: 10.3934/math.2023236

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Research article

Critical regularity of nonlinearities in semilinear effectively damped wave models

Abdelhamid Mohammed Djaouti ^{1
,
,},
Michael Reissig ²

1.
Preparatory Year Deanship, King Faisal University, Hofuf 31982, Al-Ahsa, Saudi Arabia
2.
Faculty for Mathematics and Computer Science, Technical University Bergakademie, Prüferstr. 9, Freiberg 09596, Germany

Received: 29 August 2022 Revised: 07 November 2022 Accepted: 16 November 2022 Published: 08 December 2022
MSC : 35L52, 35L71

In this paper we consider the Cauchy problem for the semilinear effectively damped wave equation

$ \begin{equation*} u_{tt}-u_{xx}+b(t)u_{t} = |u|^{3}\mu(|u|), \, \, \, u(0, x) = u_{0}(x), \, \, \, u_{t}(0, x) = u_{1}(x). \end{equation*} $

Our goal is to propose sharp conditions on $ \mu $ to obtain a threshold between global (in time) existence of small data Sobolev solutions (stability of the zero solution) and blow-up behaviour even of small data Sobolev solutions.
- damped wave equations,
- time dependent dissipation,
- global existence,
- blow-up,
- critical regularity
Citation: Abdelhamid Mohammed Djaouti, Michael Reissig. Critical regularity of nonlinearities in semilinear effectively damped wave models[J]. AIMS Mathematics, 2023, 8(2): 4764-4785. doi: 10.3934/math.2023236

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Abstract

In this paper we consider the Cauchy problem for the semilinear effectively damped wave equation

$ \begin{equation*} u_{tt}-u_{xx}+b(t)u_{t} = |u|^{3}\mu(|u|), \, \, \, u(0, x) = u_{0}(x), \, \, \, u_{t}(0, x) = u_{1}(x). \end{equation*} $

Our goal is to propose sharp conditions on $ \mu $ to obtain a threshold between global (in time) existence of small data Sobolev solutions (stability of the zero solution) and blow-up behaviour even of small data Sobolev solutions.

References

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