Abstract
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.
Highlights
Let Ω be a bounded domain in Rn with the Lipschitz continuous boundary ∂Ω
In [1,2,3], we only have that a sequence of approximate solutions to (1) converges to a function u in an appropriate function space, and that if u satisfies the energy conservation law, it is a weak solution to (1)
The limit should satisfy the energy conservation law, and existence theorem of a global weak solution has not been established yet
Summary
A nonlinear equation of motion of vibrating membrane with a “viscosity” term is investigated. The term −Δut is added, and it is well known that this equation is well posed in the space of W1,2 functions. The viscosity term is changed to −(div(∇u/√1 + |∇u|2))t, and it is proved that if initial data is slightly smooth (but belonging to W2,2 is sufficient), a weak solution exists uniquely in the space of BV functions
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