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Global smooth solutions for a class of parabolic integrodifferential equations. (English) Zbl 0848.45002
The author studies partial integrodifferential equations of the type \[ u_{tt} (x,t)- \varphi (u_x (x,t) )_x- \int^t_0 a(t- s)\psi (u_x (x,s) )_{xs} ds= f(x, t), \] for \(0< x< 1\) and \(t>0\) with zero boundary data and \(u(x, 0)\) and \(u_t (x, 0)\) given. It is shown that if \(\psi'\) is positive and bounded, \(\varphi\) is Lipschitz-continuous, and \(a\) is more singular than \(t^{- 2/3}\) near \(t=0\), then there exists a (global) solution for which \(u_{tt}\) and \(u_{xxt}\) are integrable to some high power. The uniqueness of such solutions is shown in greater generality as well as results on a class of related linear equations with continuous coefficients. These results make it possible to prove higher smoothness properties. No assumptions that the initial values or the forcing function should be small are needed.

MSC:
45K05 Integro-partial differential equations
45G10 Other nonlinear integral equations
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