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Transactions of the American Mathematical Society

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A quasilinear hyperbolic integro-differential equation related to a nonlinear string


Author: Melvin L. Heard
Journal: Trans. Amer. Math. Soc. 285 (1984), 805-823
MSC: Primary 45K05; Secondary 35L70, 73K03
DOI: https://doi.org/10.1090/S0002-9947-1984-0752504-5
MathSciNet review: 752504
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Abstract: We discuss global existence, boundedness and regularity of solutions to the integrodifferential equation

\begin{displaymath}\begin{array}{*{20}{c}} {{u_{t\,t}}\,(t,x) + c\,(t)\,{u_t}(t,... ..._t}(0,x) = {u_1}(x), \qquad x \in {\mathbf{R}}.} \\ \end{array}\end{displaymath}

This type of equation occurs in the study of the nonlinear behavior of elastic strings. We show that if the initial data $ {u_0}\,(x),{u_1}\,(x)$ is small in a suitable sense, and if the damping coefficient $ c\,(t)$ grows sufficiently fast, then the above equation possesses a globally defined classical solution for forcing terms $ h\,(t,x,u)$ which are sublinear in $ u$. In the nonlinearity we require that $ M \in {C^1}\,[0,\infty )$ and, in addition, satisfies $ M( \lambda ) \geq {m_0} > 0$ for all $ \lambda \geq 0$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0752504-5
Keywords: Integrodifferential equations, evolution equations, stable family of generators, energy estimates, elastic strings
Article copyright: © Copyright 1984 American Mathematical Society

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