A transmission problem for thermoelastic plates

Abstract

In this paper we study a transmission problem for thermoelastic plates. We prove that the problem is well-posed in the sense that there exists only one solution which is as regular as the initial data. Moreover, we prove that the local thermal effect is strong enough to produce uniform rate of decay of the solution. More precisely, there exist positive constants <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma"> <mml:semantics> <mml:mi>γ</mml:mi> <mml:annotation encoding="application/x-tex">\gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the total energy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E\left ( t \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis t right-parenthesis less-than-or-equal-to upper C upper E left-parenthesis 0 right-parenthesis e Superscript minus gamma t"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>C</mml:mi> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>γ</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E\left ( t \right ) \le CE\left ( 0 \right ){e^{ - \gamma t}}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula>.