Differential and Integral Equations

Similarity solutions for a class of hyperbolic integrodifferential equations

Hans Engler

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For a class of hyperbolic partial integrodifferential equations of the form $u_{tt} - u_{xx} + a*u_{xx} = 0$, fundamental solutions are found that depend on the similarity variable $\xi =x(t-|x|)^{-\alpha}$, where $\alpha \in (0,1)$ and the integral kernel $a$ behaves like $t^{-\alpha}$ near $t=0$. The asymptotic behavior of these solutions in various scaling limits and their regularity is discussed. Applications to solutions of general initial-value problems of such equations are given.

Article information

Differential Integral Equations, Volume 10, Number 5 (1997), 815-840.

First available in Project Euclid: 1 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]
Secondary: 35L99: None of the above, but in this section 45D05: Volterra integral equations [See also 34A12]


Engler, Hans. Similarity solutions for a class of hyperbolic integrodifferential equations. Differential Integral Equations 10 (1997), no. 5, 815--840. https://projecteuclid.org/euclid.die/1367438621

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