Mathematica Bohemica, first online, pp. 1-14


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

Aya Khaldi, Amar Ouaoua, Messaoud Maouni

Received July 8, 2020.   Published online November 16, 2021.

Abstract:  We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms $u_t-M\biggl(\int_\Omega\vert\nabla u \vert^2 {\rm d}x\bigg) \Delta u+ \vert u \vert^{m(x) -2}u_t= \vert u \vert^{r(x) -2}u.$ 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.
Keywords:  Kirchhoff equation; reaction-diffusion equation; variable exponent; global solution
Classification MSC:  35B40, 35L70, 35L10
DOI:  10.21136/MB.2021.0122-20

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Affiliations:   Aya Khaldi, Laboratory of Applied Mathematics and History and Didactics of Mathematics, Faculty of Sciences, University of Skikda, B.P. 26 Route d'El-Hadaiek, Skikda 21000, Algeria, e-mail: ayakhaldi21@gmail.com; Amar Ouaoua, Laboratory of Applied Mathematics and History and Didactics of Mathematics, Faculty of Technology, University of Skikda, B.P. 26 Route d'El-Hadaiek, Skikda 21000, Algeria, e-mail: a.ouaoua@univ-skikda.dz, ouaouaama21@gmail.com; Messaoud Maouni, Laboratory of Applied Mathematics and History and Didactics of Mathematics, Faculty of Sciences, University of Skikda, B.P. 26 Route d'El-Hadaiek, Skikda 21000, Algeria, e-mail: m.maouni@univ-skikda.dz, maouni21@gmail.com


 
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