An iteration procedure for a class of integrodifferential equations of parabolic type (Q914113)

This paper considers the initial boundary value problem \[ (1)\quad Lu=h(x,t)+\int_{\Omega}B(x,t,u,u_ x,u_{xx})dx, \] \[ u(x,t)=0,\quad (x,t)\in S_ T=\partial \Omega \times (0,T),\quad u(x,0)=u_ 0(x),\quad x\in {\bar \Omega}, \] where \(L=\partial /\partial t- [a_{ij}(x,t)\partial^ 2/\partial x_ i\partial x_ j)+b_ i(x,t)\partial /\partial x_ i+C(x,t)]\) is a parabolic operator with \(a_{ij}\xi_ i\xi_ j\geq a_ 0| \xi |^ 2\) \((a_ 0>0)\) for \(\xi \in R^ n\). These integrodifferential models take into account the effect of the past history arise in physical, engineering and biological problems (population models). The authors introduce the notion of strong solution of problem (1), and prove existence of the solution by using the Green's function along with Gronwall's inequality. Then continuous dependence of the classical solution upon known functions is shown and the uniqueness is obtained as a direct corollary of it. This paper does not contain any numerical technique for solving these kinds of integro-differential equations.