Self-Similar Blow-Up Solutions of the KPZ Equation
Self-similar blow-up solutions for the generalized deterministic KPZ equation with are considered. The asymptotic behavior of self-similar solutions is studied.
We consider the generalized deterministic KPZ equationwhere and . Equation (1) was first considered in the case by Kardar et al.  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 . 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 and initial function were proven in . This result was extended to and in  and to and in . 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 . To confirm the optimality of obtained existence conditions, the authors of  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 formAfter substitution of (2) into (1) we find thatand should satisfy the following equation:
We will add to (4) the following initial data:
Let us state the main result.
A simple computation shows that Theorem 1 is a consequence of the following statement.
2. The Proof of Theorem 2
We start with a simple result which is used later on.
Now we will obtain the upper bound for .
Lemma 4. There exists such that
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).
Remark 6. Note that Theorem 2 demonstrates the optimality of Theorem in . The arguments are the same as in Remark of that paper.
Our next result shows that (4) with initial data has no global solution.
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.
This work was supported by the State Research Program of Belarus (Grant no. 1.2.03).
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