Stability and Gevrey regularity of the semigroup associated with an Euler–Bernoulli plate equation subject to localized Kelvin–Voigt damping

Abstract We consider an Euler–Bernoulli plate equation with Kelvin–Voigt damping in a bounded domain. The damping is localized in an appropriate open strict subset $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> of the domain $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> . While it is known that the solutions of this model with a full damping $$\omega = \Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> generate an analytic semigroup, this property is no longer valid for locally distributed damping. In view of this, we study regularity of the equation as expressed by a membership in an appropriate Gevrey’s class. It turns out that the final result depends on both “geometric” and analytical properties of the support function defining the dissipation. First, assuming that the damping coefficient d is $$C^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and satisfies some structural conditions, we prove that the underlying semigroup is of Gevrey class s : for every $$s&gt;7/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>7</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , when the damping region is a collar around the whole boundary, for every $$s&gt;4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , when the damping region is more general, for every $$s&gt;7/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>7</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , when the damping region is more general, and the function d is $$C^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> and satisfies one more structural condition than in the two cases above. In all cases, the semigroup is infinitely differentiable on $$(0,\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and also exponentially stable. Next, we drop the smoothness assumption on the damping coefficient and show that the corresponding semigroup decays at a rational rate $$O(t^{-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Our proofs are based on the frequency domain method combined with an adequate smoothing procedure, interpolation inequalities, and multipliers technique. The main features of our proof of the Gevrey regularity are: (i) the introduction of suitable auxiliary functions, (ii) an appropriate estimate of the localized kinetic energy and exploitation of structural constraints on the damping coefficient to prove localized regularity, and, (iii) the use of Hahn–Banach theorem to derive a sharp estimate of a negative norm of the velocity nonsmooth component, (iv) the use of commutators to simplify our presentation. Our rational stability and Gevrey regularity results are the first for the plate equation with localized Kelvin–Voigt damping in higher space dimensions.