Abstract Partially supporte by CICYT Research Grant PB86–0112–00202 and EEC Contrast SCI–0019–c. We consider the Cauchy Problem where p > 1 and U$sub:o$esub:(x) is continuous, nonnegative and bounded. Let u(x.t) be the solution of (1). (2). and assume that u blows up at t=T . and We then show that the blow–up set is discrete. Moreover, if x=0 is a blow–up point, one of the two following possibilities occurs. Either There exist c > 0 and an even number m such that Keywords: Partially supported by CICYT Research Grant PB86-0112-C0202 and EEC Contract SCI-0019-C