# Self-Similar Blow-Up Solutions of the KPZ Equation

Authors: Alexander Gladkov
Journal: International Journal of Differential Equations
 (2015)

---

## Abstract

Self-similar blow-up solutions for the generalized deterministic KPZ equationu
t
=
u
x
x
+
|
u
x
|
q with q
>
2 are considered. The asymptotic behavior of self-similar solutions is studied.


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## Body

## 1. Introduction

We consider the generalized deterministic KPZ equation(1)∂u∂t=∂2u∂x2+∂u∂xqforx,t∈ST≔R×0,T,where q
>
2 and T
>
0. Equation (1) was first considered in the case q
=
2 by Kardar et al. [1] in connection with the study of the growth of surfaces. When q
=
2, (1) has since been referred to as the deterministic KPZ equation. For q
≠
2 it also called the generalized deterministic KPZ equation or Krug-Spohn equation because it was introduced in [2]. We refer to the review article [3] for references and a detailed historical account of the KPZ equation.The existence and uniqueness of a classical solution of the Cauchy problem for (1) with q
=
1 and initial function u
0
∈
C
0
3
(
R
n
) were proven in [4]. This result was extended to u
0
∈
C
2
(
R
n
)
∩
W
2
,
∞
(
R
n
) and q
≥
1 in [5] and to u
0
∈
C
(
R
n
)
∩
L
∞
(
R
n
) and q
≥
0 in [6]. Several papers [7–11] were devoted to the investigation of the Cauchy problem for irregular initial data, namely, for u
0
∈
L
p
(
R
n
), 1
≤
p
<
∞, or for bounded measures. The existence and uniqueness of a solution to the Cauchy problem with unbounded initial datum are proved in [12]. To confirm the optimality of obtained existence conditions, the authors of [12] analyze the asymptotic behavior of self-similar blow-up solutions of (1) for q
<
2.In this paper we investigate the asymptotic behavior of self-similar blow-up solutions of (1) with q
>
2 having the form(2)ux,t=T-tαfξ,whereξ=xT-tβ,0<t<T.After substitution of (2) into (1) we find that(3)α=q-22q-1,β=-12and f should satisfy the following equation:(4)f′′+f′q-12ξf′+αf=0on0,+∞.We will add to (4) the following initial data:(5)f0=-f0<0,f′0=0.Put(6)C=1q-1q-1qq1/q-1.Let us state the main result.Theorem 1.
Letu be a self-similar blow-up solution of (1) with q
>
2 which is defined in (2)–(5). Then(7)limt→T⁡ux,tT-t1/q-1=Cxq/q-1.A simple computation shows that Theorem1 is a consequence of the following statement.Theorem 2.
Letq
>
2 and let f be a solution of problem (4), (5). Then(8)limξ→∞⁡fξξq/q-1=C.The behavior of self-similar solutions for (1) of the type u
(
x
,
t
)
=
t
α
g
(
x
t
β
) has been analyzed in [13].


## 2. The Proof of Theorem2

We start with a simple result which is used later on.Lemma 3.
Letf be a solution of problem (4), (5) defined on [
0
,
ξ
-
). Then (9)f′ξ>0,f′′ξ>0forξ∈0,ξ-.Proof.
Obviously,f
′
′
(
0
)
=
α
f
0
>
0
. Therefore, by continuity, f
′
′
>
0 and f
′
>
0 in some right-neighborhood of 0. Suppose that there exists ξ
0 such that 0
<
ξ
0
<
ξ
-, f
′
′
>
0 on [
0
,
ξ
0
) and f
′
′
(
ξ
0
)
=
0. Then f
′
>
0 on (
0
,
ξ
0
] and f
′
′
′
(
ξ
0
)
≤
0. From (4) we find that f
′
′
′
(
ξ
0
)
=
f
′
(
ξ
0
)
/
[
2
(
q
-
1
)
]
>
0. This contradiction proves (9).Now we will obtain the upper bound forf
′.Lemma 4.
There existsξ
0
>
0 such that (10)f′ξ<ξ21/q-1forξ≥ξ0.Proof.
Lemma3 implies that f
(
ξ
)
→
∞ as ξ
→
ξ
- and that there exists unique point ξ
0
∈
(
0
,
ξ
-
) such that f
<
0 on (
0
,
ξ
0
) and f
>
0 on (
ξ
0
,
ξ
-
)
. Substituting f
′
′
>
0 and f
≥
0 in (4) yields f
′
(
ξ
)
<
{
ξ
/
2
}
1
/
(
q
-
1
) for ξ
∈
[
ξ
0
,
ξ
-
)
. Thus, ξ
-
=
∞ and (10) holds.Changing variables in (4) (11)f′ξ=ξ1/q-1gt,ξ=exp⁡t,we get the new equation (12)g′′+3-qq-1g′-q-2q-12g=12g′-gq′+1q-1g-qq-1gqexp⁡2t.By (9), (10), and (11), there hold (13)gt>0for  anyt∈R,(14)gt<121/q-1,g′t>-gq-1for large values of t. Put (15)C0=1q1/q-1,C1=12q1/q-1.It is obvious that C
0
>
C
1. Now we will establish the asymptotic behavior of g
(
t
) as t
→
+
∞
.Lemma 5.
Assume thatg is defined in (11). Then (16)limt→+∞⁡gt=C0.Proof.
From a careful inspection of (12) we conclude that a local maximum of g
(
t
) can happen only when g
(
t
)
>
C
0.
At first we suppose thatg
(
t
) does not tend to C
0 as t
→
+
∞ and g
(
t
) is monotonic solution of (12) for large values of t
. Then there exists C
-
≠
C
0 such that l
i
m
t
→
∞
g
(
t
)
=
C
-
. It is not difficult to show that for any ε
>
0 there exist A
>
0 and a sequence {
t
k
}
k
=
1
∞ with the properties: (17)limk→∞⁡tk=+∞,g′′tk≤A,g′tk≤ε.Indeed, let g
′
≥
0 for the definiteness. We suppose that g
′
(
t
) is not monotonic function for large values of t since otherwise (17) is obvious. Denote by {
τ
k
}
k
=
1
∞ a sequence of local minima for g
′. Then (17) holds for some subsequence of {
τ
k
}
k
=
1
∞.
Passing to the limit in (12) as t
=
t
k
→
+
∞ and choosing ε in a suitable way we get that the left-hand side is bounded, while the right-hand side tends to infinity if C
-
≠
0
. Let C
-
=
0. Using (13) and (14) we conclude from (12) that (18)g′′+3-qq-1g′≥g3q-1exp⁡2tfor large values of t. Then for large values of k (17) and (18) imply (19)gtk≤γexp⁡-2tk,where positive constant γ does not depend on k
. Setting ξ
k
=
e
x
p
t
k, from (11) and (19), we get (20)f′ξk≤γξk3-2q/q-1that contradicts (9).
Now until the end of the proof we assume thatg
(
t
) is not monotonic solution of (12) for large values of t
. Suppose that l
i
m
i
n
f
t
→
∞
g
(
t
)
<
C
0
. Then there exist positive unbounded increasing sequences {
s
k
}
k
=
1
∞ and {
t
k
}
k
=
1
∞ such that t
k
>
s
k, (21)g′t≤0fort∈sk,tk,and g
(
s
k
)
=
C
0, g
(
t
k
)
=
C
⋆, where C
1
<
C
⋆
<
C
0
. Then (22)12g′-gq′=-qgq-1-C1q-1g′≥-qC⋆q-1-C1q-1g′≥0onsk,tk.So, (12) and (22) imply that (23)g′′t+3-qq-1g′t≥-qC⋆q-1-C1q-1g′texp⁡2skfort∈sk,tk.Hence, integrating with respect to t from s
k to t
k, we get (24)g′t+3-qq-1gtsktk≥qC⋆q-1-C1q-1C0-C⋆exp⁡2sk.This leads to a contradiction, since (13), (14), and (21) imply that the left-hand side of the last inequality is bounded, while the right-hand side becomes unbounded as k
→
∞
.
Let us prove thatl
i
m
i
n
f
t
→
∞
g
(
t
)
=
C
0
. Indeed, otherwise, there exist ε
>
0 and a sequence {
τ
k
}
k
=
1
∞ of local minima for g with the properties τ
k
→
+
∞ as k
→
+
∞ and (25)gτk≥C0+ε.Passing in (12) to the limit as t
=
τ
k
→
+
∞ we get a contradiction.
To end the proof we show thatl
i
m
s
u
p
t
→
∞
g
(
t
)
=
C
0
. Otherwise, l
i
m
s
u
p
t
→
∞
g
(
t
)
>
C
0
. Then there exist unbounded increasing sequences {
s
k
}
k
=
1
∞ and {
t
k
}
k
=
1
∞ such that t
k
>
s
k
>
2, (26)g′sk=0,g′tk=0,g′t≥0gtk>C0+δ,gsk-C0<ε,fort∈sk,tk,where δ
>
0 and (27)ε=min⁡δ2,q-14C0δ2,1-781/q-1C0.Without loss of a generality we can suppose (28)C0-ε<gsk<C0or (29)C0≤gsk<C0+ε.Let (28) be valid. If (29) is realized, the arguments are similar and simpler. Denote by {
t
-
k
}
k
=
1
∞ a sequence such that (30)t-k∈sk,tk,gt-k=C0.Applying Hölder’s inequality we derive (31)∫t-ktkg′τdτ≤∫t-ktkg′τ2exp⁡2τdτ1/2·∫t-ktkexp⁡-2τdτ1/2and therefore (32)∫t-ktkg′τ2exp⁡2τdτ≥2δ2exp⁡2t-k.We multiply (12) by g
′
(
t
) and integrate after over [
s
k
,
t
k
]
. Using (15), (26)–(28), (30), and (32) we obtain (33)-q-22q-12g2tk≤q-3q-1∫sktkg′τ2dτ+∫sktkg′τ212-qgq-1τexp⁡2τdτ+exp⁡2t-kq-1·∫skt-k12g2τ′-qq+1gq+1τ′dτ≤-14·∫t-ktkg′τ2exp⁡2τdτ+exp⁡2t-kq-1g2τ2-qgq+1τq+1skt-k≤-δ22+εC0q-1exp⁡2t-k≤-δ24exp⁡2t-k.Passing to the limit as k
→
∞ we get a contradiction with (14).Now (8) is a simple consequence of Lemma 5 and the definition of g
(
t
).Remark 6.
Note that Theorem2 demonstrates the optimality of Theorem  2.3 in [12]. The arguments are the same as in Remark 4.6 of that paper.
Our next result shows that (4) with initial data (34)f0=f0>0,f′0=0has no global solution.Theorem 7.
Letq
>
2 and let f be a solution of problem (4), (34). Then there exists ξ
⋆ such that 0
<
ξ
⋆
<
+
∞ and f
(
ξ
)
→
-
∞ as ξ
↑
ξ
⋆.Proof.
Suppose that problem (4), (34) has a solution f that is infinitely extendible to the right. Using the arguments of Lemma 3 we show that f
′
<
0 and f
′
′
<
0 on (
0
,
+
∞
)
. From (4) we obtain (35)f′′′ξ<-f′ξq′.After the integration of (35) over [
0
,
ξ
] we conclude that (36)f′′ξ<-f′ξq.Integrating (36) over [
ξ
1
,
ξ
]
(
0
<
ξ
1
<
ξ
) we infer (37)1q-1f′ξ1q-1>ξ-ξ1.Passing to the limit as ξ
→
∞ we obtain a contradiction which proves the theorem.



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*Source: 572841-2015-08-26.xml*