×

On a class of integrodifferential equations. (English) Zbl 1127.45005

The paper deals with the following system of integro-partial differential equations:
\[ \frac{\partial u}{\partial t} + \sum_{i=1}^N A_i \partial_i u + Bu - \alpha \Delta u - \beta \int_0^t G(t-s)\Delta u(x,\, s)\, ds = f - \nabla p, \]
\[ div\, u = 0, \]
in \(\Omega\times (0,\,T)\), together with initial and boundary conditions:
\[ u(x,\,t) = 0, \text{on}\; \partial \Omega \times (0,\, T) \]
\[ u(x,\,0) = u_0. \]
Here, \(\alpha,\, \beta >0\), \(\Omega\) is a domain in \({\mathbb R}^n\), \(B\) and \(A_i\, (i=1,\dots,n)\) are \(N\times N\)-matrices, \(G\) is a kernel and \(f,\, p\) are given functions. Based on a Faedo-Galerkin approach, the existence of a solution for the variational formulation of the problem is proved. By employing supplementary hypotheses on the kernel \(G\) and \(f\), the authors show the uniqueness of the solution and obtain some regularity results.

MSC:

45K05 Integro-partial differential equations
45F05 Systems of nonsingular linear integral equations
45L05 Theoretical approximation of solutions to integral equations
PDFBibTeX XMLCite