IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi Publishing Corporation 10.1155/2015/572841 572841 Research Article Self-Similar Blow-Up Solutions of the KPZ Equation http://orcid.org/0000-0002-6255-1161 Gladkov Alexander Messaoudi Salim Department of Mathematics and Mechanics Belarusian State University Nezavisimosti Avenue 4 220030 Minsk Belarus bsu.by 2015 2682015 2015 07 07 2015 16 08 2015 2682015 2015 Copyright © 2015 Alexander Gladkov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction

We consider the generalized deterministic KPZ equation(1)ut=2ux2+uxqfor  x,tSTR×0,T,where q > 2 and T > 0 . Equation (1) was first considered in the case q = 2 by Kardar et al.  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 . We refer to the review article  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 . This result was extended to u 0 C 2 ( R n ) W 2 , ( R n ) and q 1 in  and to u 0 C ( R n ) L ( R n ) and q 0 in . Several papers  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 . To confirm the optimality of obtained existence conditions, the authors of  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,where  ξ=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+fq-12ξf+αf=0on  0,+.

We will add to (4) the following initial data:(5)f0=-f0<0,f0=0.

Put(6)C=1q-1q-1qq1/q-1.

Let us state the main result.

Theorem 1.

Let u be a self-similar blow-up solution of (1) with q > 2 which is defined in (2)–(5). Then(7)limtTux,tT-t1/q-1=Cxq/q-1.

A simple computation shows that Theorem 1 is a consequence of the following statement.

Theorem 2.

Let q > 2 and let f be a solution of problem (4), (5). Then(8)limξfξξq/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 .

2. The Proof of Theorem <xref ref-type="statement" rid="thm2">2</xref>

Lemma 3.

Let f be a solution of problem (4), (5) defined on [ 0 , ξ - ) . Then (9)fξ>0,fξ>0for  ξ0,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 for f .

Lemma 4.

There exists ξ 0 > 0 such that (10)fξ<ξ21/q-1for  ξξ0.

Proof.

Lemma 3 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)fξ=ξ1/q-1gt,ξ=expt,we get the new equation (12)g+3-qq-1g-q-2q-12g=12g-gq+1q-1g-qq-1gqexp2t.By (9), (10), and (11), there hold (13)gt>0for  any  tR,(14)gt<121/q-1,gt>-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 that g 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 that g ( 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)limktk=+,gtkA,gtkε.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-1gg3q-1exp2tfor large values of t . Then for large values of k (17) and (18) imply (19)gtkγexp-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)fξkγξk3-2q/q-1that contradicts (9).

Now until the end of the proof we assume that g ( 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)gt0for  tsk,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-qCq-1-C1q-1g0on  sk,tk.So, (12) and (22) imply that (23)gt+3-qq-1gt-qCq-1-C1q-1gtexp2skfor  tsk,tk.Hence, integrating with respect to t from s k to t k , we get (24)gt+3-qq-1gtsktkqCq-1-C1q-1C0-Cexp2sk.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 that l 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)gτkC0+ε.Passing in (12) to the limit as t = τ k + we get a contradiction.

To end the proof we show that l 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)gsk=0,gtk=0,gt0gtk>C0+δ,gsk-C0<ε,for  tsk,tk,where δ > 0 and (27)ε=minδ2,q-14C0δ2,1-781/q-1C0.Without loss of a generality we can suppose (28)C0-ε<gsk<C0or (29)C0gsk<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-ksk,tk,gt-k=C0.Applying Hölder’s inequality we derive (31)t-ktkgτdτt-ktkgτ2exp2τdτ1/2·t-ktkexp-2τdτ1/2and therefore (32)t-ktkgτ2exp2τdτ2δ2exp2t-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-12g2tkq-3q-1sktkgτ2dτ+sktkgτ212-qgq-1τexp2τdτ+exp2t-kq-1·skt-k12g2τ-qq+1gq+1τdτ-14·t-ktkgτ2exp2τdτ+exp2t-kq-1g2τ2-qgq+1τq+1skt-k-δ22+εC0q-1exp2t-k-δ24exp2t-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 Theorem 2 demonstrates the optimality of Theorem   2.3 in . 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,f0=0has no global solution.

Theorem 7.

Let q > 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)fξ<-fξq.After the integration of (35) over [ 0 , ξ ] we conclude that (36)fξ<-fξq.Integrating (36) over [ ξ 1 , ξ ] ( 0 < ξ 1 < ξ ) we infer (37)1q-1fξ1q-1>ξ-ξ1.Passing to the limit as ξ we obtain a contradiction which proves the theorem.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the State Research Program of Belarus (Grant no. 1.2.03).

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