The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces

• Published in: Journal de mathématiques pures et appliquées Vol. 81; no. 4; pp. 343 - 378
• Main Authors: Ben-Artzi, Matania, Souplet, Philippe, Weissler, Fred B.
• Format: Journal Article
• Language: English
• Published: Paris Elsevier SAS 2002; Elsevier
• Subjects: Classical and quantum physics: mechanics and fields; Classical mechanics of discrete systems: general mathematical aspects; Critical exponents; Exact sciences and technology; Lebesgue spaces; Nonexistence; Nonlinear parabolic equations; Nonuniqueness; Physics; Viscous Hamilton–Jacobi equations; Well-posedness; Nonuniqueness; Viscous Hamilton–Jacobi equations; Nonexistence; 35B33; Lebesgue spaces; 35K15; 35B40; 35K55; Critical exponents; Nonlinear parabolic equations; Well-posedness; Viscosity; Hamilton-Jacobi equations; Non uniqueness; Lebesgue space; Well posed problem; Viscous equation; Non existence
• ISSN: 0021-7824
• DOI: 10.1016/S0021-7824(01)01243-0

We consider viscous Hamilton–Jacobi equations of the form
(VHJ)
u
t−Δu=a|∇u|
p,
x∈
R
N,
t>0,
u(x,0)=u
0(x),
x∈
R
N,
where
a∈
R
,
a≠0 and
p⩾1. We provide an extensive investigation of the local Cauchy problem for (VHJ) for irregular initial data
u
0, namely for
u
0 in Lebesgue spaces
L
q=L
q(
R
N)
, 1⩽
q<∞. The case of initial data measures or in Sobolev spaces is also considered.
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 
u
0.
In the case
a>0 and
u
0⩾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
u
0⩾0 (absorbing gradient term), we show that existence holds in
L
1 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 
u
0), 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.