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Similarity solutions for a class of hyperbolic integrodifferential equations. (English) Zbl 0892.45005
The author constructs fundamental solutions to a class of equations of the form \[ u_{tt}(x,t)-u_{xx}(x,t)+\int _0^t a(t-s)u_{xx}(x,s)ds=0, \] where the kernel \(a\) behaves like \(t^{-\alpha }\) near \(t=0\) \((\alpha \in (0,1))\). These solutions depend on the similarity variable \(x(t-| x | )^{-\alpha }\). The construction relies on the solution of the Rayleigh problem. A number of equivalent representations is derived and some of their properties together with consequences for the solutions of general initial-value problems are given. Relations between the asymptotic behaviour of solutions and the kernels are also studied as well as regularity properties of solutions across \(\pm x=t,\) at \(x=0,\) and near \(t=0\).

45K05 Integro-partial differential equations
45M05 Asymptotics of solutions to integral equations