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

We study the local Cauchy problem for the semilinear parabolic equation u_t - Δu = a |∇u|^p, t > 0, x ∈ ℝ^N, with p ≥ 1, a ≠ 0, and initial data in L^q(ℝ^N), 1 ≤ q < ∞. After showing local nonexistence when p ≥ 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 ≥ q_c, and ill posed, due to nonuniqueness, if 1 ≤ q < q_c (implying, in particular, p > (N + 2)/(N + 1)). To prove nonuniqueness, for a > 0, we construct a self-similar, positive, regular solution u, such that lim_t↓0 ∥u(t)∥_L^q = 0.