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Global existence of solutions of Volterra integrodifferential equations of parabolic type. (English) Zbl 0780.45012
The paper deals with the initial-boundary value problem (1) \(u_ t(t,x)= \int_ 0^ t k(t-s)\sigma \bigl(u_ x(s,x)\bigr)_ x\text{d}s\) \(+f(t,x),\) \(t \geq 0\), \(x \in(0,1)\); \(u(0,x)= u_ 0(x)\), \(x \in[0,1]\); \(u(t,0)=u(t,1)=0\), \(t<0\), whose various variants appear in mathematical models of viscoelasticity. The author shows that if \(k\) is sufficiently close to the delta functional then the problem (1) is of parabolic type. In particular, if the function \(k \in L^ 1_{\text{loc}}(R^ +,R)\) is of positive type and there exists a constant \(\kappa>0\) such that \(\text{Re}[\hat k(z)]\geq \kappa | \text{Im}[\hat k(z)|\) holds for all \(z\) such that \(\text{Re}[z]>0\), then under some additional assumptions on \(\sigma,f\) and \(u_ 0\) the given problem possesses a strong solution.
Reviewer: M.TvrdĂ˝ (Praha)

45K05 Integro-partial differential equations
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