A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy

Mohammad Shahrouzi; Jorge Ferreıra

doi:10.33434/cams.941324

EN

A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy

Abstract

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.

Keywords

Kirchhoff equation, stability result, variable exponents, blow-up

References

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Mohammad Shahrouzi ^*
Iran

Jorge Ferreıra
Brazil

Publication Date

December 27, 2021

Submission Date

May 25, 2021

Acceptance Date

December 13, 2021

Published in Issue

Year 2021 Volume: 4 Number: 4

DOI

https://doi.org/10.33434/cams.941324

IZ

https://izlik.org/JA22JE96GX

APA

Shahrouzi, M., & Ferreıra, J. (2021). A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy. Communications in Advanced Mathematical Sciences, 4(4), 208-216. https://doi.org/10.33434/cams.941324

AMA

1.Shahrouzi M, Ferreıra J. A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy. Communications in Advanced Mathematical Sciences. 2021;4(4):208-216. doi:10.33434/cams.941324

Chicago

Shahrouzi, Mohammad, and Jorge Ferreıra. 2021. “A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions With Positive Initial Energy”. Communications in Advanced Mathematical Sciences 4 (4): 208-16. https://doi.org/10.33434/cams.941324.

EndNote

Shahrouzi M, Ferreıra J (December 1, 2021) A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy. Communications in Advanced Mathematical Sciences 4 4 208–216.

IEEE

[1]M. Shahrouzi and J. 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, vol. 4, no. 4, pp. 208–216, Dec. 2021, doi: 10.33434/cams.941324.

ISNAD

Shahrouzi, Mohammad - Ferreıra, Jorge. “A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions With Positive Initial Energy”. Communications in Advanced Mathematical Sciences 4/4 (December 1, 2021): 208-216. https://doi.org/10.33434/cams.941324.

JAMA

1.Shahrouzi M, Ferreıra J. A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy. Communications in Advanced Mathematical Sciences. 2021;4:208–216.

MLA

Shahrouzi, Mohammad, and 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, vol. 4, no. 4, Dec. 2021, pp. 208-16, doi:10.33434/cams.941324.

Vancouver

1.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. 2021 Dec. 1;4(4):208-16. doi:10.33434/cams.941324

Cited By

General decay and blow up of solutions for a plate viscoelastic p ( x )-Kirchhoff type equation with variable exponent nonlinearities and boundary feedback

Quaestiones Mathematicae

https://doi.org/10.2989/16073606.2023.2256983

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