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Volume 34, Issue 4
New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions

Hanni Dridi & Khaled Zennir

DOI: 10.4208/jpde.v34.n4.2

J. Part. Diff. Eq., 34 (2021), pp. 313-347.

Published online: 2021-08

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  • Abstract

In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows

\begin{equation*}u_{tt} -\mathcal{K}\left( \mathcal{N}u(t)\right)\left[   \Delta_{p(x)}u +\Delta_{r(x)}u_{t}\right]=\mathcal{F}(x, t, u).\end{equation*}

Here, $\mathcal{K}\left( \mathcal{N}u(t)\right)$ is a Kirchhoff function, $\Delta_{r(x)}u_{t}$ represent a Kelvin-Voigt strong  damping term, and $\mathcal{F}(x, t, u)$ is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for $p(x), q(x), r(x), \mathcal{F}(x,t,u)$ and the initial data for which the blow-up occurs.

  • Keywords

Galerkin approximation, variable exponents, Kirchhoff equation, blow-up of solutions, Kelvin-Voigt damping, general nonlinearity.

  • AMS Subject Headings

35B44, 35B40

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hannidridi@gmail.com (Hanni Dridi)

khaledzennir4@gmail.com (Khaled Zennir)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-34-313, author = {Dridi , Hanni and Zennir , Khaled}, title = {New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions}, journal = {Journal of Partial Differential Equations}, year = {2021}, volume = {34}, number = {4}, pages = {313--347}, abstract = {

In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows

\begin{equation*}u_{tt} -\mathcal{K}\left( \mathcal{N}u(t)\right)\left[   \Delta_{p(x)}u +\Delta_{r(x)}u_{t}\right]=\mathcal{F}(x, t, u).\end{equation*}

Here, $\mathcal{K}\left( \mathcal{N}u(t)\right)$ is a Kirchhoff function, $\Delta_{r(x)}u_{t}$ represent a Kelvin-Voigt strong  damping term, and $\mathcal{F}(x, t, u)$ is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for $p(x), q(x), r(x), \mathcal{F}(x,t,u)$ and the initial data for which the blow-up occurs.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v34.n4.2}, url = {http://global-sci.org/intro/article_detail/jpde/19402.html} }
TY - JOUR T1 - New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions AU - Dridi , Hanni AU - Zennir , Khaled JO - Journal of Partial Differential Equations VL - 4 SP - 313 EP - 347 PY - 2021 DA - 2021/08 SN - 34 DO - http://doi.org/10.4208/jpde.v34.n4.2 UR - https://global-sci.org/intro/article_detail/jpde/19402.html KW - Galerkin approximation, variable exponents, Kirchhoff equation, blow-up of solutions, Kelvin-Voigt damping, general nonlinearity. AB -

In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows

\begin{equation*}u_{tt} -\mathcal{K}\left( \mathcal{N}u(t)\right)\left[   \Delta_{p(x)}u +\Delta_{r(x)}u_{t}\right]=\mathcal{F}(x, t, u).\end{equation*}

Here, $\mathcal{K}\left( \mathcal{N}u(t)\right)$ is a Kirchhoff function, $\Delta_{r(x)}u_{t}$ represent a Kelvin-Voigt strong  damping term, and $\mathcal{F}(x, t, u)$ is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for $p(x), q(x), r(x), \mathcal{F}(x,t,u)$ and the initial data for which the blow-up occurs.

Hanni Dridi & Khaled Zennir. (2021). New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions. Journal of Partial Differential Equations. 34 (4). 313-347. doi:10.4208/jpde.v34.n4.2
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