The local theory for viscous hamilton-Jacobi equations in Lebesgue spaces

We consider viscous Hamilton-Jacobi equations of the form where a ε ℝ, a ≠ 0 and p ≥ 1. We provide an extensive investigation of the local Cauchy problem for (VHJ) for irregular initial data u₀, namely for u₀ in Lebesgue spaces L^q = L^q (ℝ^N), 1 ≤ q < ∞. The case of initial data measures or in Sobolev spaces is also considered. u_t - Δu = a ∇u^p, x ε ℝ^N, t > 0, u(x, 0) = u₀(x), x ε ℝ^N, When p < 2, we prove well-posedness in L^q for q ≥ q_c = N (p - 1)/(2 - p). This holds without sign restriction neither on a nor on u₀. In the case a > 0 and u₀ ≥ 0 (repulsive gradient term) we show that existence fails in all L^q spaces when p ≥ 2. When p < 2, we prove that both existence and uniqueness fail if 1 ≤ q < q_c. Rather surprisingly, in the case a < 0 and u₀ ≥ 0 (absorbing gradient term), we show that existence holds in L¹ while it may fail in measures. More precisely, we obtain existence in L^q for any q ≥ 1 when p ≤ 2 (and also for p > 2 under some additional assumption on u₀), whereas nonexistence occurs for a large class of measure initial data if p > (N+2)/(N+1). In particular, a critical exponent for existence and uniqueness in the scale of L^q spaces appears if the gradient term is repulsive, while none occurs if it is absorbing.