The Cauchy Problems For The Nonlinear Damped Timoshenko Systems And Thermoelasity Of Type Ⅲ

In this paper,we investigated the initial-boundary value problem of Timoshenko type system and thermoelasticity of type III whose source and nonlinear damping term only in one equation respectively:(1)Timoshenko type system is damped by nonlinear damping term,only in the equation for the rotation angle,no direct damping is applied on the equation for the transverse displacement of the beam.#12(2)Thermoelasticity of type III is damped by nonlinear damping term,only in the equation for temperature,no direct damping is applied on the equation for the displacement vector.#12 where σ(v)and f((?))are like functions like σ(v)=)=|v|rv,r>-1,f((?))=|(?)|α(?),α>0.We give the proofs of global existence and uniqueness of the two systems.We utilize Faedo-Galerkin method and a standard compactness argument for our purpose and for this it suffices to derive two decay estimates respectively for the two cases of r>0 and-1<r<0 by energy methods and Nakao’s lemma.