The abstract Cauchy problem for dissipative operators with respect to metric-like functionals

This paper is devoted to a characterization of semigroups of Lipschitz operators on a closed subset D of a Banach space X and the abstract Cauchy problem for an operator A in X satisfying the following condition: There exists a proper lower semicontinuous functional φ from X   into [0,∞][0,∞] such that the effective domain of φ   is D(A)D(A) and such that limn→∞⁡Axn=Axlimn→∞⁡Axn=Ax in X   for any x∈D(A)x∈D(A) and any sequence {xn}{xn} in D(A)D(A) satisfying two conditions limn→∞⁡xn=xlimn→∞⁡xn=x in X   and limsupn→∞φ(xn)≤φ(x). The main result asserts that a semigroup of Lipschitz operators on D can be generated by an operator A satisfying the above-mentioned condition, a dissipative condition with respect to a metric-like functional and a subtangential condition. The Kirchhoff equations with acoustic boundary conditions are solved by the method based on the abstract result with the construction of suitable Liapunov functionals and the use of a metric-like functional on a suitable set.