On the Reduction Procedure for a Nonlinear Integro-Differential Equation

/ is a real-valued function on Jt, and i f u dx/JG G I =f dx. Detailed assumptions on the domain 0, on the coefficients aij , and on the right-hand sides g, h are formulated in Se'ction 3. It is our aim to describe the set I of all triples (u, g, h) such that t isa solution of (1) for, the right-hand sides g, h. We apply the method of global reduction to a one-dimensional problem. Our result will be that I has the structure I = a x 2, Where a = {(s, t) E R2 I (s) = t} with a suitable chosen function T, and S is ' a linear manifold. The method of one-dimensional reduction was used by BEROER and PODOLAJ [2] and by CAFAGNA and DONATI [3]' for the solution of boundary value problems for semiinear elliptic differential operators. The differentiability of the considered operatom and the kind of interaction of the nonlinearity with the spectrum of the linear part of the operator play an important role for these considerations We can look at the problem (1) as a simplified semiinear elliptic equation. The: effect of this simplification is that (i) we need notto make any assumptions on the nonlinearity (ii) the behavior of the nonlinearity / with respect to the spectrum of the linear part does not play any role. In Section2 we present abstract results, which we will apply to th6 solution of