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In this paper, we consider a non-local diffusion equation involving thefractional $p(x)$-Laplacian with nonlinearities of variable exponent type.Employing the sub-differential approach we establish the existence of localsolutions. By combining the potential well theory with the Nehari manifold, weobtain the existence of global solutions and finite time blow-up of solutions.Moreover, we study the asymptotic stability of global solutions as time goes toinfinity in some variable exponent Lebesgue spaces.

Global existence and blow-up of solutions for a parabolic equation involving the fractional $p(x)$-Laplacian