A class of nonlinear parabolic integro-differential equations (Q1261639)

The author studies the integro-differential equation \[ u_ t- u_{xx}=a{d\over dt}\max\left(\int^ 1_ 0u(x,t)dt,0\right)+f(x,t) \] in \((0,1)\times(0,T)\) with initial and boundary conditions. For \(a<1\), an existence and uniqueness theorem is proven. A regularity result is stated. For \(a=1\), examples of the nonexistence and nonuniqueness of the problem are given. [Remark: The existence proof is given by the use of Schauder's fixed point theorem: (1) The set \(K(A,M)\), defined by equation (3.8) is not closed in \(H\) as asserted; (2) there is no proof of compactness].