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This paper is devoted to the study of the large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions. This work is the sequel of Hu et al. (SIAM J Control Optim 53:378-398, 2015) in which a probabilistic method was developed to show that the solution of a parabolic semilinear PDE behaves like a linear term $$\lambda T$$ shifted with a function v, where $$(v,\lambda )$$ is the solution of the ergodic PDE associated to the parabolic PDE. We adapt this method in finite dimension by a penalization method in order to be able to apply an important basic coupling estimate result and with the help of a regularization procedure in order to avoid the lack of regularity of the coefficients in finite dimension. The advantage of our method is that it gives an explicit rate of convergence. [ABSTRACT FROM AUTHOR]

We consider the problem of boundary feedback stabilization of a vibrating string that is fixed at one end and with control action at the other end. In contrast to previous studies that have required L²-regularity for the initial position and H-1-regularity for the initial velocity, in this paper we allow for initial positions with L¹-regularity and initial velocities in W-1,1 on the space interval. It is well known that for a certain feedback parameter, for sufficiently regular initial states the classical energy of the closed-loop system with Neumann velocity feedback is controlled to zero after a finite time that is equal to the minimal time where exact controllability holds. In this paper, we present a Dirichlet boundary feedback that yields a well-defined closed-loop system in the (L¹, W-1,1) framework and also has this property. Moreover, for all positive feedback parameters our feedback law leads to exponential decay of a suitably defined L¹-energy. For more regular initial states with (L², H-1) regularity, the proposed feedback law leads to exponential decay of an energy that corresponds to this framework. If the initial states are even more regular with H¹-regularity of the initial position and L²-regularity of the initial velocity, our feedback law also leads to exponential decay of the classical energy. [ABSTRACT FROM AUTHOR]