TY  - JOUR
ID  - Rzepnicki2013
LB  - Rzepnicki2013
AU  - Rzepnicki, Łukasz
TI  - Generating the exponentially stable C<sub>0</sub>-semigroup in a nonhomogeneous string equation with damping at the end
ST  - Generating the exponentially stable C<sub>0</sub>-semigroup in a nonhomogeneous string equation with damping at the end
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2013
VL  - 33
IS  - 1
SP  - 151
EP  - 162
DO  - http://dx.doi.org/10.7494/OpMath.2013.33.1.151
UR  - http://dx.doi.org/10.7494/OpMath.2013.33.1.151
AB  - 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. \(x \in [0, 1]\).
KW  - nonhomogeneous string
KW  - one-dimensional wave equation
KW  - exponentially stable \(C_0\)-semigroup
KW  - Hilbert space
SN  - 1232-9274
ER  -