On the energy decay rate of the fractional wave equation on ℝ with relatively dense damping
Walton Green
Abstract
We establish upper bounds for the decay rate of the energy of the damped fractional wave equation when the averages of the damping coefficient on all intervals of a fixed length are bounded below. If the power of the fractional Laplacian, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is between 0 and 2, the decay is polynomial. For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s \ge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the decay is exponential. Our assumption is also necessary for energy decay. Second, we prove that exponential decay cannot hold for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than 2"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if the damping vanishes at all.