TY - JOUR
T1 - The local theory for viscous hamilton-Jacobi equations in Lebesgue spaces
AU - Ben-Artzi, Matania
AU - Souplet, Philippe
AU - Weissler, Fred B.
PY - 2002
Y1 - 2002
N2 - 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 Lq = Lq (ℝN), 1 ≤ q < ∞. The case of initial data measures or in Sobolev spaces is also considered. ut - Δu = a ∇up, x ε ℝN, t > 0, u(x, 0) = u0(x), x ε ℝN, When p < 2, we prove well-posedness in Lq for q ≥ qc = 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 Lq spaces when p ≥ 2. When p < 2, we prove that both existence and uniqueness fail if 1 ≤ q < qc. 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 Lq 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 Lq spaces appears if the gradient term is repulsive, while none occurs if it is absorbing.
AB - 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 Lq = Lq (ℝN), 1 ≤ q < ∞. The case of initial data measures or in Sobolev spaces is also considered. ut - Δu = a ∇up, x ε ℝN, t > 0, u(x, 0) = u0(x), x ε ℝN, When p < 2, we prove well-posedness in Lq for q ≥ qc = 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 Lq spaces when p ≥ 2. When p < 2, we prove that both existence and uniqueness fail if 1 ≤ q < qc. 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 Lq 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 Lq spaces appears if the gradient term is repulsive, while none occurs if it is absorbing.
KW - Critical exponents
KW - Lebesgue spaces
KW - Nonexistence
KW - Nonlinear parabolic equations
KW - Nonuniqueness
KW - Viscous Hamilton-Jacobi equations
KW - Well-posedness
UR - http://www.scopus.com/inward/record.url?scp=0036077877&partnerID=8YFLogxK
U2 - 10.1016/S0021-7824(01)01243-0
DO - 10.1016/S0021-7824(01)01243-0
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AN - SCOPUS:0036077877
SN - 0021-7824
VL - 81
SP - 343
EP - 378
JO - Journal des Mathematiques Pures et Appliquees
JF - Journal des Mathematiques Pures et Appliquees
IS - 4
ER -