Existence and blow-up of logarithmic-viscoelastic wave equation with fractional Kirchhoff type and variables exponents

Mama Saouli; Mohamed  Saadaoui

doi:10.5269/bspm.76202

Mama Saouli University of Laghouat
Mohamed Saadaoui

Resumo

In this paper, we investigate the existence of solutions for a Viscoelastic Wave equation with fractional kirchhof f type and logarithmic term and variables exponents. In the initial boundary value problem of this work, we focus on using the Gagliardo seminorm [.]_α,2, and the fractional Laplace operator (-∆)α where 0 < α < 1. Firstly, we prove the existence of the weak solutions, concerning this issue under suitable assumptions on the variables exponents of nonlinear source term, we use the Galerkin’s approximation method. Subsequently, we establish the long-time behavior of solutions with nonpositive initial energy.

Downloads

Não há dados estatísticos.

Referências

P. Hartman Ordinary Differential Equations, Birkhäuser, Boston, Mass, (1982).

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, (1997).

G. Autuori, P. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 489-516 , 196 (2010).

E.Pi.skin, Blow up of solutions for a nonlinear viscoalastic wave equations with variable exponents. Middle East Journal of Science, 5(2), 134-145, (2019).

Q. Lin, X. Tian, R. Xu and M. Zhang, Blow up and blow up time for degenerate Kirchhoff type wave problems involving the fractional Laplacian with arbitrary positive initial energy. Discrete Contin. Dyn. Syst. Ser. S, 13, 2095-2107, (2020).

Tae Gab Hal , Sun-Hye Park, Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity exponents. Ha and Park Advances in Difference Equations, 235, (2020).

Mingqi Xiang, B. Zhang and D. Hu, Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping. Electron. Res. Arch., 28, 651-669, (2020).

M. Shahrouzi, Kargarfard, Firoozeh, Blow-up of solutions for a Kirchhoff type equation with variable-exponent nonlinearities . Journal of Applied Analysis, 27(1), 97-105, (2021).

Mingqi Xiang, Die Hu, Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity. Discrete and Continuous Dynamical Systems Series. 14(12), 4609-4629, (2021).

M. Xiang, D. Hu, B. Zhang and Y. Wang, Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growth. J. Math. Anal. Appl., 501, 19pp, (2021).

Abita Rahmoune, Logarithmic wave equation involving variable-exponents nonlinearities : Well-posedness and blw-up. WSEAS TRANSACTIONS on MATHEMATICS, 2224-2880, (2022).

Y. Wu, Y, Gao, Blow up of solutions of Kirchhoff type viscoelastic wave equations with logarithmic nonlinearity of variable exponents. Research Square, (2023).

F. Demengel, G. Demengel, Espaces Fonctionnels Utilisation dans la r´esolution des ´equations aux d´eriv´ees partielles, EDP Sciences, CNRS ´EDITIONS.

D. Edmunds, W.D. Evans, 230 FRACTIONAL SOBOLEV SPACES AND INEQUALITIES, CAMBRIDGE UNIVERSITY PRESS, (2023).

R. Aounallah, A. Choucha, S. Boulaaras, Asymptotic behavior of a logarithmic-viscoelastic wave equation with internal fractional damping. Periodica Mathematica Hungarica, Springer, 10, (2024).

L. Zhang, Y. Liu, A Class of Fractional Viscoelastic Kirchhoff Equations Involving Two Nonlinear Source Terms of Different Signs. Axioms, 169, 13(3), 2075-1680, (2024).