Self-Similar Blow-Up Solutions of the KPZ Equation

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).