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Keywords:
Kirchhoff equation; reaction-diffusion equation; variable exponent; global solution
Kirchhoff equation; reaction-diffusion equation; variable exponent; global solution
Summary:
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.
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