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.

Full text (pdf)

Opuscula Mathematica - cover

Cite this article as:
Łukasz Rzepnicki, Generating the exponentially stable C0-semigroup in a nonhomogeneous string equation with damping at the end, Opuscula Math. 33, no. 1 (2013), 151-162, http://dx.doi.org/10.7494/OpMath.2013.33.1.151

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.