Energy conservative solutions to the system of full variational sine-Gordon equations in a unit sphere (Q2795513)

The authors consider a hysteretic reaction-diffusion system subject to Neumann boundary condition and periodic conditions on a bounded domain in \(\mathbb R^n\) with smooth boundary. For the biological diffusion model considered, the hysteretic effect is described by the generalized play operator and thus uniqueness of periodic solutions is still left. In fact, this system with the mentioned conditions can be regarded as a sort of quasi-variational evolution inequalities.NEWLINENEWLINENEWLINETo overcome the difficulties due to the quasi-variational structure, the authors started by analyzing an approximation problem, which is obtained from the considered problem by applying a truncation operator to nonlinear terms and adding a diffusion term in the equation. Then with the help of the Poincaré map, it is used the Leray-Schauder theorem in order to prove the existence of a solution to the approximation problem. Finally, based on the uniform bounds for the last approximation solution it is proved that the initial problem admits a solution, which is the limit of some subsequence of approximate solutions.