On a class of nonlocal parabolic equations

We consider a boundary value problem for parabolic equations with nonlocal nonlinearity of such a form that favorably differs from other equations in that it leads to partial differential equations that have important properties of ordinary differential equations. Local solvability and uniqueness theorems are proved, and an analog of the Painlevé singular nonfixed points theorem is proved. In this case, there is an alternative – either a solution exists for all $t\ge0$ or it goes to infinity in a finite time $t=T$ (blowup mode). Sufficient conditions for the existence of a blowup mode are given.