International Journal of Differential Equations
Volume 2015 (2015), Article ID 572841, 4 pages
http://dx.doi.org/10.1155/2015/572841

Research Article

Self-Similar Blow-Up Solutions of the KPZ Equation

Alexander Gladkov

Department of Mathematics and Mechanics, Belarusian State University, Nezavisimosti Avenue 4, 220030 Minsk, Belarus

Received 7 July 2015; Accepted 16 August 2015

Academic Editor: Salim Messaoudi

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.

Abstract.

Self-similar blow-up solutions for the generalized deterministic KPZ equation with are considered. The asymptotic behavior of self-similar solutions is studied.

1. Introduction

We consider the generalized deterministic KPZ equation

where and . Equation (1) was first considered in the case by Kardar et al. [1] in connection with the study of the growth of surfaces. When , (1) has since been referred to as the deterministic KPZ equation. For 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 and initial function were proven in [4]. This result was extended to and in [5] and to and in [6]. Several papers [7–11] were devoted to the investigation of the Cauchy problem for irregular initial data, namely, for , , 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 .

In this paper we investigate the asymptotic behavior of self-similar blow-up solutions of (1) with having the form

After substitution of (2) into (1) we find that

and should satisfy the following equation:

We will add to (4) the following initial data:

Put

Let us state the main result.

Theorem 1. Let be a self-similar blow-up solution of (1) with which is defined in (2)–(5). Then

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

Theorem 2. Let and let be a solution of problem (4), (5). Then

The behavior of self-similar solutions for (1) of the type has been analyzed in [13].

2. The Proof of Theorem 2

We start with a simple result which is used later on.

Lemma 3. Let be a solution of problem (4), (5) defined on . Then

Proof. Obviously, Therefore, by continuity, and in some right-neighborhood of . Suppose that there exists such that , on and . Then on and . From (4) we find that . This contradiction proves (9).

Now we will obtain the upper bound for .

Lemma 4. There exists such that

Proof. Lemma 3 implies that as and that there exists unique point such that on and on Substituting and in (4) yields for Thus, and (10) holds.

Changing variables in (4)

we get the new equation

By (9), (10), and (11), there hold

for large values of . Put

It is obvious that . Now we will establish the asymptotic behavior of as

Lemma 5. Assume that is defined in (11). Then

Proof. From a careful inspection of (12) we conclude that a local maximum of can happen only when .
At first we suppose that does not tend to as and is monotonic solution of (12) for large values of Then there exists such that It is not difficult to show that for any there exist and a sequence with the properties:

Indeed, let for the definiteness. We suppose that is not monotonic function for large values of since otherwise (17) is obvious. Denote by a sequence of local minima for . Then (17) holds for some subsequence of .
Passing to the limit in (12) as and choosing in a suitable way we get that the left-hand side is bounded, while the right-hand side tends to infinity if Let . Using (13) and (14) we conclude from (12) that

for large values of . Then for large values of (17) and (18) imply

where positive constant does not depend on Setting , from (11) and (19), we get

that contradicts (9).
Now until the end of the proof we assume that is not monotonic solution of (12) for large values of Suppose that Then there exist positive unbounded increasing sequences and such that ,

and , , where Then

So, (12) and (22) imply that

Hence, integrating with respect to from to , we get

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
Let us prove that Indeed, otherwise, there exist and a sequence of local minima for with the properties as and

Passing in (12) to the limit as we get a contradiction.
To end the proof we show that Otherwise, Then there exist unbounded increasing sequences and such that ,

where and

Without loss of a generality we can suppose

or

Let (28) be valid. If (29) is realized, the arguments are similar and simpler. Denote by a sequence such that

Applying Hölder’s inequality we derive

and therefore

We multiply (12) by and integrate after over Using (15), (26)–(28), (30), and (32) we obtain

Passing to the limit as we get a contradiction with (14).

Now (8) is a simple consequence of Lemma 5 and the definition of .

Remark 6. Note that Theorem 2 demonstrates the optimality of Theorem   in [12]. The arguments are the same as in Remark of that paper.
Our next result shows that (4) with initial data

has no global solution.

Theorem 7. Let and let be a solution of problem (4), (34). Then there exists such that and as .

Proof. Suppose that problem (4), (34) has a solution that is infinitely extendible to the right. Using the arguments of Lemma 3 we show that and on From (4) we obtain

After the integration of (35) over we conclude that

Integrating (36) over we infer

Passing to the limit as 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).

References

M. Kardar, G. Parisi, and Y.-C. Zhang, “Dynamic scaling of growing interfaces,” Physical Review Letters, vol. 56, no. 9, pp. 889–892, 1986.

J. Krug and H. Spohn, “Kinetic roughening of growing surfaces,” in Solids Far from Equilibrium, C. Godrèche, Ed., pp. 479–582, Cambridge University Press, Cambridge, UK, 1991.

I. Corwin, “The Kardar-Parisi-Zhang equation and universality class,” Random Matrices: Theory and Applications, vol. 1, Article ID 1130001, p. 76, 2012.

M. Ben-Artzi, “Global existence and decay for a nonlinear parabolic equation,” Nonlinear Analysis, vol. 19, no. 8, pp. 763–768, 1992.

L. Amour and M. Ben-Artzi, “Global existence and decay for viscous Hamilton-Jacobi equations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 31, no. 5-6, pp. 621–628, 1998.

B. H. Gilding, M. Guedda, and R. Kersner, “The Cauchy problem for ,” Journal of Mathematical Analysis and Applications, vol. 284, no. 2, pp. 733–755, 2003.

S. Benachour and P. Laurençot, “Global solutions to viscous Hamilton-Jacobi equations with irregular initial data,” Communications in Partial Differential Equations, vol. 24, no. 11-12, pp. 1999–2021, 1999.

S. Benachour, B. Roynette, and P. Vallois, “Solutions fondamentales de ,” Astérisque, vol. 236, pp. 41–71, 1996.

M. Ben-Artzi, J. Goodman, and A. Levy, “Remarks on a nonlinear parabolic equation,” Transactions of the American Mathematical Society, vol. 352, no. 8, pp. 731–751, 2000.

M. Ben-Artzi, P. Souplet, and F. B. Weissler, “Sur la non-existence et la non-unicité des solutions du problème de Cauchy pour une équation parabolique semi-linéaire,” Comptes Rendus de l'Académie des Sciences Series I: Mathematics, vol. 329, no. 5, pp. 371–376, 1999.

M. Ben-Artzi, P. Souplet, and F. B. Weissler, “The local theory for viscous hamilton-Jacobi equations in Lebesgue spaces,” Journal des Mathematiques Pures et Appliquees, vol. 81, no. 4, pp. 343–378, 2002.

A. Gladkov, M. Guedda, and R. Kersner, “A KPZ growth model with possibly unbounded data: correctness and blow-up,” Nonlinear Analysis, Theory, Methods and Applications, vol. 68, no. 7, pp. 2079–2091, 2008.

M. Guedda and R. Kersner, “Self-similar solutions to the generalized deterministic KPZ equation,” Nonlinear Differential Equations and Applications, vol. 10, no. 1, pp. 1–13, 2003.