Email this article Logarithmic character of large time asymptotics for solutions of Sobolev type nonlinear equations with cubic nonlinearity
- Authors: Naumkin P.I.1
-
Affiliations:
- National Autonomous University of Mexico, Center of Mathematical Sciences
- Issue: Vol 214, No 7 (2023)
- Pages: 134-160
- Section: Articles
- URL: https://journal-vniispk.ru/0368-8666/article/view/133543
- DOI: https://doi.org/10.4213/sm9515
- ID: 133543
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Abstract
The Cauchy problem of the form
{i∂t(u−∂2xu)+∂2xu−a∂4xu=u3,t>0, x∈R,u(0,x)=u0(x),x∈R,
is considered for a Sobolev-type nonlinear equation with cubic nonlinearity, where a>1/5, a≠1. It is shown that the asymptotic behaviour of the solution is characterized by an additional logarithmic decay in comparison with the corresponding linear case. To find the asymptotics of solutions of the Cauchy problem for a nonlinear Sobolev-type equation, factorization technique is developed. To obtain estimates for derivatives of the defect operators, L2-estimates of pseudodifferential operators are used.
About the authors
Pavel Ivanovich Naumkin
National Autonomous University of Mexico, Center of Mathematical Sciences
Author for correspondence.
Email: pavelni@matmor.unam.mx
References
- С. Л. Соболев, “Об одной новой задаче математической физики”, Изв. АН СССР. Сер. матем., 18:1 (1954), 3–50
- А. Г. Свешников, А. Б. Альшин, М. О. Корпусов, Ю. Д. Плетнер, Линейные и нелинейные уравнения соболевского типа, Физматлит, М., 2007, 734 с.
- A. B. Al'shin, M. O. Korpusov, A. G. Sveshnikov, Blow-up in nonlinear Sobolev type equations, De Gruyter Ser. Nonlinear Anal. Appl., 15, Walter de Gruyter & Co., Berlin, 2011, xii+648 pp.
- M. O. Korpusov, A. V. Ovchinnikov, A. G. Sveshnikov, E. V. Yushkov, Blow-up in nonlinear equations of mathematical physics. Theory and methods, De Gruyter Ser. Nonlinear Anal. Appl., 27, De Gruyter, Berlin, 2018, xviii+326 pp.
- Е. И. Кайкина, П. И. Наумкин, И. А. Шишмарев, “Задача Коши для уравнения типа Соболева со степенной нелинейностью”, Изв. РАН. Сер. матем., 69:1 (2005), 61–114
- Е. И. Кайкина, П. И. Наумкин, И. А. Шишмарев, “Асимптотика решений при больших временах для нелинейных уравнений типа Соболева”, УМН, 64:3(387) (2009), 3–72
- N. Hayashi, P. I. Naumkin, “Nongauge invariant cubic nonlinear Schrödinger equations”, Pac. J. Appl. Math. Yearbook, 1 (2012), 1–16
- P. I. Naumkin, I. Sanchez-Suarez, “On the critical nongauge invariant nonlinear Schrödinger equation”, Discrete Contin. Dyn. Syst., 30:3 (2011), 807–834
- N. Hayashi, P. I. Naumkin, “Global existence for the cubic nonlinear Schrodinger equation in low order Sobolev spaces”, Differential Integral Equations, 24:9-10 (2011), 801–828
- N. Hayashi, T. Ozawa, “Scattering theory in the weighted $L^{2}(mathbb R^{n})$ spaces for some Schrödinger equations”, Ann. Inst. H. Poincare Phys. Theor., 48:1 (1988), 17–37
- N. Hayashi, P. I. Naumkin, “The initial value problem for the cubic nonlinear Klein–Gordon equation”, Z. Angew. Math. Phys., 59:6 (2008), 1002–1028
- N. Hayashi, P. I. Naumkin, “Factorization technique for the modified Korteweg–de Vries equation”, SUT J. Math., 52:1 (2016), 49–95
- П. И. Наумкин, “Асимптотика решений модифицированного уравнения Уизема, учитывающего поверхностное натяжение”, Изв. РАН. Сер. матем., 83:2 (2019), 174–203
- П. И. Наумкин, “Оценки убывания решений задачи Коши для модифицированного уравнения Кавахары”, Матем. сб., 210:5 (2019), 72–108
- М. В. Федорюк, Асимптотика. Интегралы и ряды, Наука, М., 1987, 544 с.
- A. P. Calderon, R. Vaillancourt, “A class of bounded pseudo-differential operators”, Proc. Nat. Acad. Sci. U.S.A., 69:5 (1972), 1185–1187
- R. R. Coifman, Y. Meyer, Au delà des operateurs pseudo-differentiels, Asterisque, 57, Soc. Math. France, Paris, 1978, i+185 pp.
- H. O. Cordes, “On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators”, J. Funct. Anal., 18:2 (1975), 115–131
- I. L. Hwang, “The $L^{2}$-boundedness of pseudodifferential operators”, Trans. Amer. Math. Soc., 302, no. 1, 1987, 55–76
- T. Cazenave, Semilinear Schrödinger equations, Courant Lect. Notes Math., 10, New York Univ., Courant Inst. Math. Sci., New York; Amer. Math. Soc., Providence, RI, 2003, xiv+323 pp.
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