Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities

Abstract

Synopsis We consider the problem where H stands for the maximal monotone graph associated with the Heaviside step function. It is shown that the problem possesses at least one (strong) solution belonging to an appropriate function space. Moreover, we prove: (i) There is a smooth initial function u 0 , u 0 ≧1, where the equality holds at exactly one point such that there are at least two different solutions corresponding to the initial data u 0 . (ii) The comparison principle: The relation for any x ≠ 0, implies u 1 (t) > u 2 (t) , t ≧0 for any u 1 , u 2 solving the problem with the initial data , respectively. (iii) For a “reasonable” set of initial data the solution is uniquely determined. Moreover, the free boundary {( x, t )| u(x, t) = 1} is regular and on its complement the equation holds in a classical sense.