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Nonlinear functional integrodifferential equations in Hilbert space. (English) Zbl 0949.35140
Summary: Let \(X\) be a Hilbert space and let \(\Omega\subset\mathbb{R}^n\) be a bounded domain with smooth boundary \(\partial\Omega\). We establish the existence and norm estimation of solutions for the parabolic partial functional integro-differential equation in \(X\) \[ \begin{split}{\partial u\over\partial t}={\mathcal A}_0u(t, x)+{\mathcal A}_1u(t- h,x)+ \int^0_{-h} a(s){\mathcal A}_2u(t+ s,x) ds+\\ \int^t_0 \{k(t, s)G(s, u(s- h),x)+ H(t, s,u(s- h,x))\} dx+\\ F(t, u(t- h,x))+ f(t,x),\quad 0< t\leq T,\;x\in\Omega,\end{split} \] by using the fundamental solution.
35R10 Partial functional-differential equations
45K05 Integro-partial differential equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J60 Nonlinear elliptic equations
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