Convergence to equilibrium for some nonlinear evolution equations with dynamical boundary condition

The asymptotic behavior of global solutions of nonlinear evolution equations, in particular convergence to a certain equilibrium as time tends to infinity has become one of the main concerns in the field of nonlinear evolution equations since 1980s. For one space dimension case, significant progresses have been made (see e.g., [6]). However, the situation in higher space dimension case is much more complicated. There are counterexamples (e.g., [7]) showing that even if the nonlinearity of a semilinear parabolic equation belongs to C, the ω-limit set of its bounded global solution could be diffeomorphic to the unit circle S. Many assumptions have been made to ensure the convergence for bounded global solutions in the literature. In 1983, L. Simon [10] made the breakthrough that if the nonlinearity of a nonlinear parabolic equation is analytic in the dependent variable, then the convergence holds. His idea relies on the extension of a gradient inequality by S. Łojasiewicz for analytic functions defined in R (Ref. [5]) to the infinite-dimensional space. Since then, a lot of work has been done in this direction (see e.g., [1–4, 9, 13] and the references therein). However, most of the previous work is concerned with evolution equations subject to homogeneous boundary conditions. For many nonlinear evolution equations with other type boundary conditions (for instance, the dynamical boundary condition) which are very important from the physical point of view, the framework in the previous literature as well as the Łojasiewicz-Simon inequality which plays a crucial role cannot apply directly. In this short note we present some results on the study of convergence of global solutions for some nonlinear evolution equations with dynamical boundary condition. Under the basic assumption that the nonlinearity is analytic with respect to the dependent variable, we develop several new Łojasiewicz-Simon type inequalities (with boundary term) which vary from problem to problem (Ref. [11, 12, 14–16]) and obtain the convergence result as well as the estimates for convergence rate.