Stability of delay evolution equations with stochastic perturbations
In order to study the problem of exponential stability in the second moment of the strong solution of the time-varying delay evolution equation \[ du(t)=(A(t, u(t)) +F(u(t-h(t))))dt + B(t, u(t-\tau(t)))dW(t) \] with initial data \(u(s)=\psi(s)\) for \(s\in [-h, 0]\) in the framework of a Gelfand triple \(U\subset H=H^*\subset U^*\), the authors aim to explain two methods for the construction of a Lyapunov functional. They first remark a theorem about mean-square stability of the trivial solution of the general functional evolution equation \[ du(t)=(A(t, u(t)) + f(t, u_t))dt + B(t, u_t)dW(t) \] with initial data \(u(s)=\psi(s)\) for \(s\in [-h, 0]\) in which \(A(t,\cdot): U\to U^*\) is a family of nonlinear monotone and coercive operators. Here, of course, the concept of solution is weaker than the one introduced in similar previous papers. Then, after a long calculation, the authors obtain a sufficient condition in terms of constants and coefficients appearing in the above mentioned delay equation for the stability of the trivial solution. Finally, they apply the results of the paper to stochastic two-dimensional Navier-Stokes equations with delay. The paper concludes with a comparison between the two methods by analyzing the stability region for the parameters involved in three different reaction-diffusion equations.
scientific article; zbMATH DE number 6377309
scientific article