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Zbl 0848.45002
Engler, Hans
Global smooth solutions for a class of parabolic integrodifferential equations. (English)
[J] Trans. Am. Math. Soc. 348, No.1, 267-290 (1996). ISSN 0002-9947; ISSN 1088-6850/e

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.
[G.Gripenberg (Helsinki)]

MSC 2000:: *45K05 Integro-partial differential equations
45G10 Nonsingular nonlinear integral equations

Keywords: smooth solutions; parabolic integrodifferential equations; quasilinear; regular solution; global solution; Volterra equation

Cited in: Zbl 1035.45005

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