Об Условиях Мгновенной Компактификации Носителя Решения И О Точных Оценках Носителя В Задаче Коши Для Параболического Уравнения С Двойной Нелинейностью И Абсорбцией

Дегтярев, Сергей Петрович; Degtyarev, S. P.

doi:10.4213/sm3859

Матем. сб.

2008

DOI: 10.4213/sm3859

|View full text |Cite

Об Условиях Мгновенной Компактификации Носителя Решения И О Точных Оценках Носителя В Задаче Коши Для Параболического Уравнения С Двойной Нелинейностью И Абсорбцией

Сергей Петрович Дегтярев¹,

S. P. Degtyarev²

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2009

2021

Publication Types

Select...

Article3

Relationship

Self Cite0

Independent3

Authors

Journals

Cited by 3 publications

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

On the instantaneous shrinking of the support of a solution to the Cauchy problem for an anisotropic parabolic equation

Degtyarev

2009

Ukr Math J

View full text Add to dashboard Cite

917.956.45 We study the phenomenon of instantaneous shrinking of the support of solution to the Cauchy problem for the parabolic equation with anisotropic degeneration, double nonlinearity, and strong absorption. In terms of the behavior of locally integrable initial data, we formulate necessary and sufficient conditions for the realization of instantaneous shrinking and establish the exact (in order) bilateral estimates for the size of the support of solution. Statement of the Problem and Main Resultwhere N is the dimension of the space R N , T > 0, we consider the following Cauchy problem for the unknown function u(x, t):where β > 0, p i > 0, i = 1, N, and λ > 0 are given constants and |u 0 | β−1 u 0 (x) is a given initial locally integrable function. We consider the case of slow diffusion in all directions and strong adsorption, which imposes the following restrictions on the parameters of the problem:In the case of isotropic equation, i.e., for p i = p, i = 1, N, it is known that, under certain conditions imposed on the initial function |u 0 | β−1 u 0 (x), problem (1.1), (1.2) is solvable in a weak sense and the phenomenon of instantaneous shrinking of the support is observed (see [1][2][3][4][5][6][7][8][9][10]). This phenomenon can de described as follows: the support of the solution becomes compact at any arbitrarily small time t > 0 and shrinks for small t despite the fact that the support of the initial function coincides with entire R N . Here, we do not analyze the history of the problem in detail (see [1-10]) but note, in particular, that the following exact (in order) bilateral estimate for the size D(t) of the support of solution to problem (1.1), (1.2) was obtained for the case of isotropic equation in [10]:where D(t) = inf r : u(x, t) ≡ 0, |x| > r ,

show abstract

On the instantaneous shrinking of the support of a solution to the Cauchy problem for an anisotropic parabolic equation

Degtyarev

2009

Ukr Math J

View full text Add to dashboard Cite

show abstract

On the phenomenon of the support shrinking of a solution with a time delay and on the extinction of the solution

Degtyarev¹

2021

Sb. Math.

View full text Add to dashboard Cite

The phenomenon of support shrinking with a time delay for the solution of a doubly nonlinear degenerate parabolic equation is studied in the case of slow diffusion and strong absorption. For a nonnegative solution, a sufficient condition for support shrinking beginning with some moment of time is deduced in terms of the local behaviour of the mass of the initial datum. It is also proved that the solution vanishes identically in finite time. Bibliography: 21 titles.

show abstract

On the phenomenon of the support shrinking of a solution with a time delay and on the extinction of the solution

Дегтярев¹,

Degtyarev

2021

Математический сборник

View full text Add to dashboard Cite

Изучается явление компактификации носителя с задержкой по времени для решения параболического вырождающегося уравнения с двойной нелинейностью в случае медленной диффузии и сильной абсорбции. В терминах локального поведения массы начальных данных для неотрицательного решения получено достаточное условие компактификации носителя начиная с некоторого момента времени. Доказано также тождественное обращение решения в нуль за конечное время. Библиография: 21 название.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Cited by 3 publications

References 22 publications

On the instantaneous shrinking of the support of a solution to the Cauchy problem for an anisotropic parabolic equation

On the instantaneous shrinking of the support of a solution to the Cauchy problem for an anisotropic parabolic equation

On the phenomenon of the support shrinking of a solution with a time delay and on the extinction of the solution

On the phenomenon of the support shrinking of a solution with a time delay and on the extinction of the solution

Contact Info

Product

Resources

About