Existence and Uniqueness of a Solution in the Space of BV Functions to the Equation of a Vibrating Membrane with a “Viscosity” Term

A nonlinear equation of motion of vibrating membrane with a “viscosity” term is investigated. Usually, the term is added, and it is well known that this equation is well posed in the space of functions. In this paper, the viscosity term is changed to , and it is proved that if initial data is slightly smooth (but belonging to is sufficient), then a weak solution exists uniquely in the space of BV functions. 1. Introduction Let be a bounded domain in with the Lipschitz continuous boundary . In [1] and in the author’s previous works [2–4], the following: is investigated, which is in these works referred to as the equation of motion of vibrating membrane. Up to now, neither existence nor uniqueness of a solution to (1) is obtained. In [1–3], we only have that a sequence of approximate solutions to (1) converges to a function in an appropriate function space, and that if satisfies the energy conservation law, it is a weak solution to (1). In [1], approximate solutions are constructed by the Ritz-Galerkin method and in [2, 3] by Rothe’s method. In [2], the boundary condition is not essentially discussed, and the observation is added in [3]. In these works, the limit should satisfy the energy conservation law, and existence theorem of a global weak solution has not been established yet. Instead, in [4], linear approximation for (1) is established. On the other hand, the equation with the strong viscosity term is investigated by several authors. For example, in [5], it is investigated in the context of control theory, and it is asserted that if and , there exists a unique solution for each . Namely, the equation with strong viscosity term is well posed in , and since is a smaller class than the space of BV functions, this suggests that the influence of the term is too strong. In this paper, replacing the strong viscosity term with , we investigate it in the space of BV functions. Namely, our problem of this paper is as follows: with initial and boundary conditions We should note that the term “viscosity” probably means implying regularity. However, in this paper, we only investigate existence and uniqueness of (2)–(4), regularity is not investigated. This is the reason that in the title there is a quotation mark. A function is said to be a function of bounded variation or a BV function in if the distributional derivative is an valued finite Radon measure in . The vector space of all functions of bounded variation in is denoted by . It is a Banach space equipped with the norm (see, e.g., [6–8]). We should note that, for , the operator is multivalued. It is