Abstract Volterra integrodifferential equations with applications to parabolic models with memory. (English) Zbl 1386.35420
The paper is concerned with the integrodifferential problem
\[
\begin{cases} u'=Au+\displaystyle\int_0^t g(t-s,u(s))\,ds+f(t,u(t)),\,\,t>0,\\ u(0)=u_0\in D(A), \end{cases}{(1)}
\]
where \(A:D(A)\subset X_0\to X_0\) is a linear operator such that \(-A\) is a sectorial operator, \(X_0\) is a Banach space, and \(g\) and \(f\) are functions satisfying some assumptions. The authors investigate the existence, uniqueness, regularity, continuous dependence on the initial data, existence of a unique continuation and a blow-up alternative for an \(\varepsilon\)-regular mild solution of \((1)\). Some applications of the obtained results to Navier-Stokes equations with memory, reaction-diffusion equations with memory, and a strongly damped plate equation with memory are finally presented.
Reviewer: Rodica Luca (Iaşi)
MSC:
| 35R09 | Integro-partial differential equations |
| 45D05 | Volterra integral equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
Keywords:
Volterra integrodifferential equation; \(\epsilon\)-regular mild solutions; existence; regularity; continuous dependence; parabolic models with memoryReferences:
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