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We consider a singularly perturbed parabolic problem with a small parameter $ \varepsilon>0 $. This problem can be regarded as an approximation of the motion of a hypersurface by its mean curvature with a driving force. In this paper we derive a rate of convergence of an order $ \varepsilon^2 $ for the motion of a smooth and compact hypersurface by its mean curvature with a driving force. We also consider the special case of a circle evolving by its curvature and show that our rate is optimal.
Leizarowitz and Mizel (1989) studied a class of one-dimensional infinite horizon variational problems arising in continuum mechanics and established that these problems possess periodic solutions. They considered a one-parameter family of integrands and show the existence of a constant $c$ such that if a parameter is larger than or equal to $c$, then the corresponding variational problem has a solution which is a constant function, while if a parameter is less than $c$, then the corresponding variational problem possesses only non-constant periodic solutions. In this paper we generalize this result for a large class of families of integrands.
We consider a coupled system of Kuramoto--Sivashinsky (KS) equations in a bounded interval depending on a suitable parameter $\nu > 0$. As $\nu$ tends to zero, we obtain a coupled system of Korteweg--de Vries (KdV) equations known to describe strong interactions of two long internal gravity waves in a stratified fluid. Existence and uniqueness of global solutions of the KS model is established as well.
Linear elliptic boundary-value problems with a parameter are studied. The Agranovich-Vishik method and specially introduced function spaces allow us to consider mixed-order problems in unbounded domains. We obtain a priori estimates and unique solvability for large values of the parameter. These results are used to study analytic semi-groups and Fredholm property of general elliptic problems in unbounded domains.