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@article{349629, author = {Pavlačková, Martina and Taddei, Valentina}, article_number = {126}, doi = {http://dx.doi.org/10.1016/j.cnsns.2023.107497}, keywords = {Damped vibrating string equation; Second-order Cauchy problem; Banach spaces; Fundamental systém; Approximation solvability method; Mild solution}, language = {eng}, issn = {1007-5704}, journal = {COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION}, title = {The damped vibrating string equation on the positive half-line}, url = {https://www.sciencedirect.com/science/article/pii/S1007570423004185}, volume = {Neuveden}, year = {2023} }
TY - JOUR ID - 349629 AU - Pavlačková, Martina - Taddei, Valentina PY - 2023 TI - The damped vibrating string equation on the positive half-line JF - COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION VL - Neuveden IS - 126 SP - 1-18 EP - 1-18 SN - 10075704 KW - Damped vibrating string equation KW - Second-order Cauchy problem KW - Banach spaces KW - Fundamental systém KW - Approximation solvability method KW - Mild solution UR - https://www.sciencedirect.com/science/article/pii/S1007570423004185 N2 - In this paper, the existence of a solution to the problem describing the small vertical vibration of an elastic string on the positive half-line is investigated in the case when both viscous and material damping coefficients are present. The result is obtained by transforming the original partial differential equation into an appropriate abstract second-order ordinary differential equation in a suitable infinite dimensional space. The abstract problem is then studied using the combination of the Kakutani fixed point theorem together with the approximation solvability method and the weak topology. The applied procedure enables obtaining the existence result also for problems depending on the first derivative, without any strict compactness assumptions put on the righthand side and on the fundamental system generated by the linear term. The paper ends by applying the obtained result to the studied mathematical model describing the small vertical vibration of an elastic string with a nonlinear Balakrishnan–Taylor-type damping term. ER -
PAVLAČKOVÁ, Martina and Valentina TADDEI. The damped vibrating string equation on the positive half-line. \textit{COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION}. 2023, Neuveden, No~126, p.~1-18. ISSN~1007-5704. Available from: https://dx.doi.org/10.1016/j.cnsns.2023.107497.
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