On nonexistence and nonuniqueness of solutions of the Cauchy problem for asemilinear parabolic equation

We study the local Cauchy problem for the semilinear parabolic equation u(t ) - Delta u = a /del u/(p), t > 0, x is an element of R-N, with p greater t han or equal to 1, a not equal 0, and initial data in L-q(R-N), 1 less than or equal to q < infinity. After showing local nonexistence when p greater than or equal to 2, we establish the existence of a critical exponent q(c) = N(p - 1)/(2 - p) for p < 2, such that the problem is well posed in L-q if q greater than or equal to q(c), and ill posed, due to nonuniqueness, if 1 less than or equal to q < q(c) (implying, in particular, p > (N + 2)/(N 1)). To prove nonuniqueness, for a > 0, we construct a self-similar, positi ve, regular solution u, such: that lim(t down arrow 0) //u(t)//(Lq) = 0. (C ) 1999 Academie des Sciences/Editions scientifiques et medicales Elsevier S AS.