













•
u
t
+ H(u)
x
= 0 (x, t) ∈ R × (0, ∞).
•
v
t
+ H(v
x
) = 0 (x, t) ∈ R × (0, ∞),
v(x, 0) = v
0
(x) x ∈ R.



t
+ H(u)ϕ
x
)dxdt +

∞
−∞
u
0
(x)ϕ(x, 0)dx = 0.
ϕ(x, t) ∈ C
1
0
(R × [0, ∞))







u C
1
u

∞
−∞

∞
0








C
σ u
l
, u
r
, H(u
l
), H(u
r
)
C

[H(u)]

= H(u
l
) − H(u
r
) σ

[u]

=

(R × [0, ∞)), ϕ ≥ 0 k ∈ R







R×(0, T ]

∞
−∞

T
0
(|u − k|ϕ
t
+ sign(u − k)(H(u) − H(k))ϕ
x
)dxdt+
+

∞
−∞
|u
0
(x) − k|ϕ(x, 0)dx −

∞
−∞

1
u
u(x, t) ∈ L
∞
(Ω × (0, ∞))
u







L
ϕ
(u) =

Ω

T
0
[|u − k|ϕ
t
+ sign(u − k)(H(u) − H(k)).∇
x
ϕ]dxdt−
−

∂Ω


u ϕ ∈ C
∞

R × (0, ∞)

u − ϕ (x
0
, t
0
) ∈ R × (0, ∞)
ϕ
t
(x
0
, t
0
) + H(ϕ
x
(x
0
, t
0
)) ≤ 0,
u ϕ ∈ C
∞

R × (0, ∞)






{u
n
(x)}
n
(a, b) C > 0
n
||u
n
||
L
∞
≤ C T V (u
n
) ≤ C
{u
n
k
(x)} {u
n
(x)}
n
u
n
k
(x) → u(x) hi k → ∞, ∀x ∈ (a, b).
u (a, b)
T V (u) ≤ lim inf
n→∞

1
≤ C|t − s|, ∀t, s ∈ [0, ∞)
{u
n
k
(x, t)} {u
n
(x, t)}
n
u
n
k
(x, t) → u(x, t) hi k → ∞, ∀(x, t) ∈ (a, b) × [0, ∞);
u
n
k
(., t) → u(., t) rong L
1
loc
(a, b) hi k → ∞, ∀t ∈ [0, ∞).
u x
(a, b)
||u(., t)||
L
∞
≤ C; T V (u(., t)) ≤ C, ∀t ∈ [0, ∞)
||u(., t) − u(., s)||
L
1
≤ C|t − s|, ∀t, s ∈ [0, ∞).

0
(x) =

u
l
x < 0,
u
r
x ≥ 0.
H
∗
H u
l
u
r
H
∗
(u; u
l
, u
r
) := sup
g

g(u)| g(u) g(u) ≤ H(u), ∀u u
l
u
r

.

, u
r
) u
l
< u
r
,
H
∗
(u; u
l
, u
r
) u
l
> u
r
.







H

H
u
1

r
σ
0
= −∞, σ
N
= +∞
σ
i
=
∼
H(u
i+1
) −
∼
H(u
i
)
u
i+1
− u
i

=
H(u
i+1
) − H(u
i
)
u
i+1

u
t
+ H(u)
x
= 0, (x, t) ∈ R × (0, T ]; u(x, 0) = u
0
(x), x ∈ R,
u
0







u(x, t)
(x, t)
u
0
H
t > 0
u(x, t)
||H||
Lip
H





(x) x ∈ R,
H [m, M]
L u
0
L
1
(R) ∩ BV (R)
[m, M] u
0
H







u
u(x, t)
||u(., t)||
∞
≤ ||u
0
||
∞
t > 0
T V (u(., t)) ≤ T V (u
0
) t > 0
||u(., t) − u(., s)||

(t) t ∈ (0, T ),
u(b, t) = u
b
(t) t ∈ (0, T ),
H u
0
, u
a
u
b
(a, b)







sign(γu(a, t) − k)(H(γu(a, t)) − H(k)) ≤ 0, ∀k ∈ I(u
a
(t), γu(a, t)),
sign(γu(b, t) − k)(H(γu(b, t)) − H(k)) ≥ 0, ∀k ∈ I(u
b
(t), γu(b, t)).
u
a
(t), u
b
(t)
u

0
(a
+
) x > a
(a, ∞).
x = a

u(x, 0) = u
0
(b
−
) x < b
u(b, t) = u
b
t ∈ (0, T ).







x = b
u(x, 0) =

u
0
(b
−
) x < b




u
t
+ H(u)
x
= 0 (x, t) ∈ (a, b) × (0, T ),
u(x, 0) = u
0
(x) x ∈ (a, b),
u(a, t) = u
a
(t) t ∈ (0, T )
u(b, t) = u
b
(t) t ∈ (0, T )
u(t, x)
t
x u(x, t)
{u
0
(x), u
a
(t), u
b
(t)} ∪ { H}.
u
u x = a, b,
||u(x, t)||