Global existence and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable exponents

Aya Khaldi; Amar Ouaoua; Messaoud Maouni

Mathematica Bohemica (2022)

  • Volume: 147, Issue: 4, page 471-484
  • ISSN: 0862-7959

Abstract

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We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms u t - M ∫ Ω | ∇ u | 2 d x Δ u + | u | m ( x ) - 2 u t = | u | r ( x ) - 2 u . We prove with suitable assumptions on the variable exponents r ( · ) , m ( · ) the global existence of the solution and a stability result using potential and Nihari’s functionals with small positive initial energy, the stability being based on Komornik’s inequality.

How to cite

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Khaldi, Aya, Ouaoua, Amar, and Maouni, Messaoud. "Global existence and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable exponents." Mathematica Bohemica 147.4 (2022): 471-484. <http://eudml.org/doc/298710>.

@article{Khaldi2022,
abstract = {We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms \begin\{equation*\} u\_\{t\}-M\biggl (\int \_\Omega \vert \nabla u \vert ^\{2\} \{\rm d\}x\bigg ) \Delta u+ \vert u \vert ^\{m(x) -2\}u\_\{t\}= \vert u \vert ^\{r(x) -2\}u. \end\{equation*\} We prove with suitable assumptions on the variable exponents $r( \{\cdot \}),$$m(\{\cdot \})$ the global existence of the solution and a stability result using potential and Nihari’s functionals with small positive initial energy, the stability being based on Komornik’s inequality.},
author = {Khaldi, Aya, Ouaoua, Amar, Maouni, Messaoud},
journal = {Mathematica Bohemica},
keywords = {Kirchhoff equation; reaction-diffusion equation; variable exponent; global solution},
language = {eng},
number = {4},
pages = {471-484},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global existence and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable exponents},
url = {http://eudml.org/doc/298710},
volume = {147},
year = {2022},
}

TY - JOUR
AU - Khaldi, Aya
AU - Ouaoua, Amar
AU - Maouni, Messaoud
TI - Global existence and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable exponents
JO - Mathematica Bohemica
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 147
IS - 4
SP - 471
EP - 484
AB - We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms \begin{equation*} u_{t}-M\biggl (\int _\Omega \vert \nabla u \vert ^{2} {\rm d}x\bigg ) \Delta u+ \vert u \vert ^{m(x) -2}u_{t}= \vert u \vert ^{r(x) -2}u. \end{equation*} We prove with suitable assumptions on the variable exponents $r( {\cdot }),$$m({\cdot })$ the global existence of the solution and a stability result using potential and Nihari’s functionals with small positive initial energy, the stability being based on Komornik’s inequality.
LA - eng
KW - Kirchhoff equation; reaction-diffusion equation; variable exponent; global solution
UR - http://eudml.org/doc/298710
ER -

References

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