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

New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions
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Year:    2021

Author:    Hanni Dridi, Khaled Zennir

Journal of Partial Differential Equations, Vol. 34 (2021), Iss. 4 : pp. 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

utt−K(Nu(t))[Δp(x)u+Δr(x)ut]=F(x,t,u).

Here, K(Nu(t)) is a Kirchhoff function, Δr(x)ut represent a Kelvin-Voigt strong  damping term, and 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),F(x,t,u) and the initial data for which the blow-up occurs.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jpde.v34.n4.2

Journal of Partial Differential Equations, Vol. 34 (2021), Iss. 4 : pp. 313–347

Published online:    2021-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    35

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

Author Details

Hanni Dridi

Khaled Zennir

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