具有临界参数的复Ginzburg-Landau方程爆破解的构造

IF 2 4区数学 Q1 MATHEMATICS

Memoirs of the American Mathematical Society Pub Date : 2019-12-10 DOI:10.1090/memo/1411

G. K. Duong, N. Nouaili, H. Zaag

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引用次数: 7

摘要

构造了一般临界情况下的复Ginzburg-Landau (CGL)方程的解，该方程在有限时间内只在一个爆炸点爆炸。我们还对其轮廓进行了清晰的描述。在第一部分中，我们正式构造了一个放大解。第二部分给出了严格的证明。证明依赖于将问题简化为有限维问题，并利用指标论得出结论。用爆破点和爆破时间来解释有限维问题的参数，可以证明构造解的稳定性。我们想提到的是，我们的解的渐近轮廓不同于以前已知的CGL或半线性热方程的轮廓。

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Construction of Blowup Solutions for the Complex Ginzburg-Landau Equation with Critical Parameters

We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time T T only at one blow-up point. We also give a sharp description of its profile. In the first part, we formally construct a blow-up solution. In the second part we give the rigorous proof. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution. We would like to mention that the asymptotic profile of our solution is different from previously known profiles for CGL or for the semilinear heat equation.

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来源期刊

Memoirs of the American Mathematical Society 数学-数学

CiteScore

3.50

自引率

5.30%

发文量

审稿时长

>12 weeks

期刊介绍： Memoirs of the American Mathematical Society is devoted to the publication of research in all areas of pure and applied mathematics. The Memoirs is designed particularly to publish long papers or groups of cognate papers in book form, and is under the supervision of the Editorial Committee of the AMS journal Transactions of the AMS. To be accepted by the editorial board, manuscripts must be correct, new, and significant. Further, they must be well written and of interest to a substantial number of mathematicians.