Observation problems posed for the Klein-Gordon equation
Transversal vibrations $u=u(x,t)$ of a string of length $l$ with fixed ends are considered, where $u$ is governed by the Klein-Gordon equation $$u_{tt}(x,t) = a^2u_{xx}(x,t)+cu(x,t), \qquad (x,t) \in [0,l] \times \mathbb{R}, \quad a>0, \ c<0.$$ Sufficient conditions are obtained that guarantee...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Szeged
2012-01-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=771 |
| Summary: | Transversal vibrations $u=u(x,t)$ of a string of length $l$ with fixed ends are considered, where $u$ is governed by the Klein-Gordon equation
$$u_{tt}(x,t) = a^2u_{xx}(x,t)+cu(x,t), \qquad (x,t) \in [0,l] \times \mathbb{R}, \quad a>0, \ c<0.$$
Sufficient conditions are obtained that guarantee the solvability of each of four observation problems with given state functions $f, \ g$ at two distinct time instants $-\infty<t_1<t_2 < \infty$. The essential conditions are the following: smoothness of $f, \ g$ as elements of a corresponding subspace $D^{s+i}(0,l)$ (introduced in [2]) of a Sobolev space $H^{s+i} (0,l)$, where $i=1,2$ depending on the type of the observation problem, and the representability of $t_2-t_1$ as a rational multiple of $\frac{2l}{a}$. The reconstruction of the unknown initial data $(u(x,0), u_t(x,0))$ as the elements of $D^{s+1}(0,l) \times D^s(0,l)$ are given by means of the method of Fourier expansions. |
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| ISSN: | 1417-3875 1417-3875 |