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On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity. (English) Zbl 0788.35001
We prove the global in time existence of smooth solutions to a nonlinear wave equation with linear viscosity describing the motion of a fixed membrane with viscosity for any smooth initial data. In our approach, the key is to prove the local in time existence of smooth solutions to the nonlinear wave equation with linear viscosity of the form: $$u_{tt}- A(u)-Bu_ t=0$$, in a bounded or unbounded domain with zero Dirichlet condition, where $$A(u)$$ is a quasilinear operator of second order and $$B$$ is a linear elliptic operator of second order. To solve the problem, we can not use the fractional power of the operator $$B$$ with zero Dirichlet condition, because its domain is not clear. We use the following simple linearization procedure. Differentiation in the time variable $$t$$ yields the equation $$v_{tt}-A'(u) v-Bv_ t=0$$ for $$v=u_ t$$. This is combined with the equation $$v_ t-A(u)- Bu_ t=0$$. Applying $$L^ p$$ theories of linear elliptic equations and linear parabolic equations to the operator $$B$$ and to the operator $$\partial_ t-B$$, respectively, and using the usual successive approximation, our local in time existence theorem is proved.

##### MSC:
 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35L70 Second-order nonlinear hyperbolic equations 74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) 35L20 Initial-boundary value problems for second-order hyperbolic equations
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