In this paper we consider the problem of robust stability for a nonlinear system of equations in partial derivatives of the reaction-diffusion type. An undisturbed system is considered to have a global attractor. The main task is to estimate the deviation of the trajectory of the perturbed system from the global attractor of the perturbed system depending on the magnitude of the perturbations. Such an estimate can be obtained in the framework of the theory of input-to-state stability (ISS). The paper does not impose any conditions on the derivative of the nonlinear interaction function, so the unity of the solution of the initial problem is not ensured. The paper proposes a new approach to obtaining estimates of robust stability of the attractor in the case of a multivalued evolutionary decoupling operator. In particular, it is proved that the multivalued decoupling operator generated by weak solutions of a nonlinear reaction-diffusion system has the property of asymptotic gain (AG) with respect to the attractor of the undisturbed system.
The paper investigates the issue of stability with respect to external disturbances for the global attractor of the wave equation under conditions that do not ensure the uniqueness of the solution to the initial problem. Under general conditions for nonlinear terms, it is proved that the global attractor of the undisturbed problem is locally stable in the sense of ISS and has the AG property with respect to disturbances.
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