Opuscula Math. 33, no. 1 (2013), 151-162
http://dx.doi.org/10.7494/OpMath.2013.33.1.151

Opuscula Mathematica

# Generating the exponentially stable C0-semigroup in a nonhomogeneous string equation with damping at the end

Łukasz Rzepnicki

Abstract. Small vibrations of a nonhomogeneous string of length one with left end fixed and right one moving with damping are described by the one-dimensional wave equation $\begin{cases} v_{tt}(x,t) - \frac{1}{\rho}v_{xx}(x,t) = 0, x \in [0,1], t \in [0, \infty),\\ v(0,t) = 0, v_x(1,t) + hv_t(1,t) = 0, \\ v(x,0) = v_0(x), v_t(x,0) = v_1(x),\end{cases}$ where $$\rho$$ is the density of the string and $$h$$ is a complex parameter. This equation can be rewritten in an operator form as an abstract Cauchy problem for the closed, densely defined operator $$B$$ acting on a certain energy space $$H$$. It is proven that the operator $$B$$ generates the exponentially stable $$C_0$$-semigroup of contractions in the space $$H$$ under assumptions that $$\text{Re}\; h \gt 0$$ and the density function is of bounded variation satisfying $$0 \lt m \leq \rho(x)$$ for a.e. $$x \in [0, 1]$$.

Keywords: nonhomogeneous string, one-dimensional wave equation, exponentially stable $$C_0$$-semigroup, Hilbert space.

Mathematics Subject Classification: 34L99, 47B44, 47D03.

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• Łukasz Rzepnicki
• Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100 Torun, Poland
• Revised: 2012-06-22.
• Accepted: 2012-07-18.