Critical regularity of nonlinearities in semilinear effectively damped wave models
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 ob...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2023-01-01
|
| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023236?viewType=HTML |
| _version_ | 1797948339640074240 |
|---|---|
| author | Abdelhamid Mohammed Djaouti Michael Reissig |
| author_facet | Abdelhamid Mohammed Djaouti Michael Reissig |
| author_sort | Abdelhamid Mohammed Djaouti |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-10T21:41:48Z |
| format | Article |
| id | doaj.art-b7104294300e4a788fda81e6ce241442 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-10T21:41:48Z |
| publishDate | 2023-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-b7104294300e4a788fda81e6ce2414422023-01-19T01:43:47ZengAIMS PressAIMS Mathematics2473-69882023-01-01824764478510.3934/math.2023236Critical regularity of nonlinearities in semilinear effectively damped wave modelsAbdelhamid Mohammed Djaouti0Michael Reissig 11. Preparatory Year Deanship, King Faisal University, Hofuf 31982, Al-Ahsa, Saudi Arabia2. Faculty for Mathematics and Computer Science, Technical University Bergakademie, Prüferstr. 9, Freiberg 09596, GermanyIn 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.https://www.aimspress.com/article/doi/10.3934/math.2023236?viewType=HTMLdamped wave equationstime dependent dissipationglobal existenceblow-upcritical regularity |
| spellingShingle | Abdelhamid Mohammed Djaouti Michael Reissig Critical regularity of nonlinearities in semilinear effectively damped wave models AIMS Mathematics damped wave equations time dependent dissipation global existence blow-up critical regularity |
| title | Critical regularity of nonlinearities in semilinear effectively damped wave models |
| title_full | Critical regularity of nonlinearities in semilinear effectively damped wave models |
| title_fullStr | Critical regularity of nonlinearities in semilinear effectively damped wave models |
| title_full_unstemmed | Critical regularity of nonlinearities in semilinear effectively damped wave models |
| title_short | Critical regularity of nonlinearities in semilinear effectively damped wave models |
| title_sort | critical regularity of nonlinearities in semilinear effectively damped wave models |
| topic | damped wave equations time dependent dissipation global existence blow-up critical regularity |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023236?viewType=HTML |
| work_keys_str_mv | AT abdelhamidmohammeddjaouti criticalregularityofnonlinearitiesinsemilineareffectivelydampedwavemodels AT michaelreissig criticalregularityofnonlinearitiesinsemilineareffectivelydampedwavemodels |