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

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.

Keywords

• kirchhoff equation
• reaction-diffusion equation
• variable exponent
• global solution