Bifurcation Analysis of a Heat Conduction Equation with Memory Effect

In this paper, we have taken into account a model that incorporates two memory integrals with weak memory kernel and is delimited by a boundary consisting of Dirichlet boundary and no-flux boundary. We prove the existence of the global attractor and obtain conditions for the global stability of trivial steady-state via constructing the Lyapunov functional. Subsequently, the stability of the trivial steady-state solutions and associated bifurcation for this model are extensively investigated. It turns out that, for small parameter, the solutions of the equation will exhibit various dynamical behaviors when the parameters R and are altered. For the inhomogeneous steady-state solution bifurcated from the trivial solution, we also provided the conditions for its stability and instability.