Abstract
In our paper we first define the mixed initial-boundary values problem in the theory of strain gradient thermoelasticity. With the help of an identity of Lagrange’s type, we then prove some theorems regarding the uniqueness of the solution of this mixed problem and also two results regarding the continuous dependence of solutions on initial data and on the charges. We must ouline that we obtain these qualitative results without recourse to any laws of conservation of energy and without recourse to any boundedness assumptions on the coefficients. It is equally important to note that we do not impose restrictions on the elastic coefficients regarding their positive definition.
Highlights
We want to underline that in the constitutive equations in the strain gradient thermoelasticity theory it is contain the second order gradient, along with the first gradient, because both have contributions to dissipation
There are many papers in the theory of elasticity of in the theory of thermoelasticity dedicated to the uniqueness of solutions or/and to continuous dependence results, but we need to say that these results are based almost exclusively on the hypothesis that the tensors of the thermoelastic coefficients are positive definite
From the studies dedicated to Cesaro means, to uniqueness and to continuous dependence results, we remember [19-21]
Summary
We want to underline that in the constitutive equations in the strain gradient thermoelasticity theory it is contain the second order gradient, along with the first gradient, because both have contributions to dissipation. In our study we address the mixed problem in the context of strain gradient thermoelasticity in a new manner, namely our approach is based on the identity of Lagrange. Other continuous dependence of solutions result it is obtained, with respect to initial data.
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