A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy
In this paper we consider $r(x)-$Kirchhoff type equation with variable-exponent nonlinearity of the form $$ u_{tt}-\Delta u-\big(a+b\int_{\Omega}\frac{1}{r(x)}|\nabla u|^{r(x)}dx\big)\Delta_{r(x)}u+\beta u_{t}=|u|^{p(x)-2}u, $$ associated with initial and Dirichlet boundary conditions. Under appropr...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Emrah Evren KARA
2021-12-01
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| Series: | Communications in Advanced Mathematical Sciences |
| Subjects: | |
| Online Access: | https://dergipark.org.tr/tr/download/article-file/1783056 |
| _version_ | 1797294231361945600 |
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| author | Mohammad Shahrouzi Jorge Ferreıra |
| author_facet | Mohammad Shahrouzi Jorge Ferreıra |
| author_sort | Mohammad Shahrouzi |
| collection | DOAJ |
| description | In this paper we consider $r(x)-$Kirchhoff type equation with variable-exponent nonlinearity of the form $$ u_{tt}-\Delta u-\big(a+b\int_{\Omega}\frac{1}{r(x)}|\nabla u|^{r(x)}dx\big)\Delta_{r(x)}u+\beta u_{t}=|u|^{p(x)-2}u, $$ associated with initial and Dirichlet boundary conditions. Under appropriate conditions on $r(.)$ and $p(.)$, stability result along the solution energy is proved. It is also shown that regarding arbitrary positive initial energy and suitable range of variable exponents, solutions blow-up in a finite time. |
| first_indexed | 2024-03-07T21:27:15Z |
| format | Article |
| id | doaj.art-d40dedd77f60436bb0e585a1e6324469 |
| institution | Directory Open Access Journal |
| issn | 2651-4001 |
| language | English |
| last_indexed | 2024-03-07T21:27:15Z |
| publishDate | 2021-12-01 |
| publisher | Emrah Evren KARA |
| record_format | Article |
| series | Communications in Advanced Mathematical Sciences |
| spelling | doaj.art-d40dedd77f60436bb0e585a1e63244692024-02-27T04:36:36ZengEmrah Evren KARACommunications in Advanced Mathematical Sciences2651-40012021-12-014420821610.33434/cams.9413241225A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial EnergyMohammad Shahrouzi0Jorge Ferreıra1Jahrom UniversityFederal University FluminenseIn this paper we consider $r(x)-$Kirchhoff type equation with variable-exponent nonlinearity of the form $$ u_{tt}-\Delta u-\big(a+b\int_{\Omega}\frac{1}{r(x)}|\nabla u|^{r(x)}dx\big)\Delta_{r(x)}u+\beta u_{t}=|u|^{p(x)-2}u, $$ associated with initial and Dirichlet boundary conditions. Under appropriate conditions on $r(.)$ and $p(.)$, stability result along the solution energy is proved. It is also shown that regarding arbitrary positive initial energy and suitable range of variable exponents, solutions blow-up in a finite time.https://dergipark.org.tr/tr/download/article-file/1783056kirchhoff equationstability resultvariable exponentsblow-up |
| spellingShingle | Mohammad Shahrouzi Jorge Ferreıra A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy Communications in Advanced Mathematical Sciences kirchhoff equation stability result variable exponents blow-up |
| title | A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy |
| title_full | A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy |
| title_fullStr | A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy |
| title_full_unstemmed | A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy |
| title_short | A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy |
| title_sort | nonlinear r x kirchhoff type hyperbolic equation stability result and blow up of solutions with positive initial energy |
| topic | kirchhoff equation stability result variable exponents blow-up |
| url | https://dergipark.org.tr/tr/download/article-file/1783056 |
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