In this paper, the focus is on the boundary stability of a nanolayer in diffusion-reaction systems, taking into account a nonlinear boundary control condition. The authors focus on demonstrating the boundary stability...In this paper, the focus is on the boundary stability of a nanolayer in diffusion-reaction systems, taking into account a nonlinear boundary control condition. The authors focus on demonstrating the boundary stability of a nanolayer using the Lyapunov function approach, while making certain regularity assumptions and imposing appropriate control conditions. In addition, the stability analysis is extended to more complex systems by studying the limit problem with interface conditions using the epi-convergence approach. The results obtained in this article are then tested numerically to validate the theoretical conclusions.展开更多
We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy...We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-Ventcel type. Under suitable and natural assumptions on the nonlinearity, we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p 〈 5. The results obtained in R^3 and RN by B. Dehman, G. Lebeau and E. Zuazua on the inequalities of the classical energy （which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball） and on Strichartz＇s estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of R^N with a subcritical nonlinearity on the domain and its boundary, and conditions on the boundary of Cauchy-Ventcel type.展开更多
文摘In this paper, the focus is on the boundary stability of a nanolayer in diffusion-reaction systems, taking into account a nonlinear boundary control condition. The authors focus on demonstrating the boundary stability of a nanolayer using the Lyapunov function approach, while making certain regularity assumptions and imposing appropriate control conditions. In addition, the stability analysis is extended to more complex systems by studying the limit problem with interface conditions using the epi-convergence approach. The results obtained in this article are then tested numerically to validate the theoretical conclusions.
文摘We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-Ventcel type. Under suitable and natural assumptions on the nonlinearity, we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p 〈 5. The results obtained in R^3 and RN by B. Dehman, G. Lebeau and E. Zuazua on the inequalities of the classical energy （which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball） and on Strichartz＇s estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of R^N with a subcritical nonlinearity on the domain and its boundary, and conditions on the boundary of Cauchy-Ventcel type.
