GREINER OPERATÖRÜYLE İLİŞKİLENDİRİLMİŞ HIZLI DİFÜZYON DENKLEMİ VE BAZI İNTEGRAL EŞİTSİZLİKLERİ

Year 2024, , 14 - 26, 26.06.2024

Ahmet Uğur Utku , Abdullah Yener

https://doi.org/10.55071/ticaretfbd.1391212

Abstract

Bu makalenin öncelikli amacı, Greiner vektör alanlarıyla ilişkilendirilmiş Carnot-Carathéodory uzayındaki düzgün sınırlara sahip sınırlı bir Ω bölgesinde,
Abstract
𝜕𝑢
!𝜕𝑡 =∇! ∙(𝐴(𝜔)∇!𝑢 )+𝑉(𝜔)𝑢 ,
𝑢(𝜔, 𝑡) = 0,
𝑢(𝜔,0) = 𝑢$(𝜔) ≥ 0,
𝜕𝑢 " " 𝜕𝑡 =∇! ∙(𝐴(𝜔)∇!𝑢 )+𝑉(𝜔)𝑢
!𝑢(𝜔,𝑡)=0
𝑢(𝜔,0) = 𝑢$(𝜔) ≥ 0
in 𝛺×(0,𝑇),
on 𝜕𝛺×(0,𝑇), in 𝛺.
""
𝛺×(0,𝑇),
𝜕𝛺 × (0, 𝑇), 𝛺
doğrusal olmayan parabolik probleminin pozitif çözümünün
( yokluğunun araştırılmasıdır. Burada, 0 < 𝑚 < 1, 𝑉 ∈ 𝐿%&'(𝛺) ve
(
0 < 𝐴(𝜔) ∈ 𝐿%&'(𝛺)’dır.
Bu makalenin diğer bir amacı ise Greiner operatörü ile ilişkilendirilmiş radyal olmayan ağırlıklı bazı Hardy tipi eşitsizlikler elde etmektir.

Keywords

Greiner operatör, Hardy tipi eşitsizlikler, hızlı difüzyon denklemi, pozitif çözümün yokluğu.

FAST DIFFUSION EQUATIONS AND SOME INTEGRAL INEQUALITIES RELATED TO GREINER OPERATOR

Abstract

The primary goal of this article is to investigate nonexistence of positive solutions to the following nonlinear parabolic problem:𝜕𝑢 " " 𝜕𝑡 =∇! ∙(𝐴(𝜔)∇!𝑢 )+𝑉(𝜔)𝑢
!𝑢(𝜔,𝑡)=0
𝑢(𝜔,0) = 𝑢$(𝜔) ≥ 0
in 𝛺×(0,𝑇),
on 𝜕𝛺×(0,𝑇), in 𝛺.
Here, 0<𝑚<1, 𝑉∈𝐿%&'(𝛺), 0<𝐴(𝜔)∈ 𝐿%&'(𝛺)andΩ is a
bounded domain with smooth boundary in Carnot-Carathéodory space related with the Greiner vector fields.
Another aim of this article is to obtain some Hardy type inequalities with non-radial weights related to Greiner operator.

Keywords

Fast diffusion equation, Greiner operator, Hardy type inequalities, nonexistence of positive solutions.

References

• Ahmetolan, S. & Kombe, I. (2016). Hardy and Rellich type inequalities with two weight functions. Mathematical Inequalities & Applications, 19(3), 937-948.
• Baras, P. & Goldstein, J. A. (1984). The heat equation with a singular potential. Transactions of the American Mathematical Society, 284(1), 121–139.
• Beals, R. & Greiner, P. C. (1988). Calculus on Heisenberg Manifolds. Annals Mathematics Studies, 119, Princeton University Press.
• Cabré, X. & Martel, Y. (1999). Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier. Comptes Rendus de l’Académié des Sciences-Series I-Mathematics, 329(11), 973–978.
• Crespo, J. A. & Alonso, I. P. (2000). Global behavior of the Cauchy problem for some critical nonlinear parabolic equations. SIAM Journal on Mathematical Analysis, 31(6), 1270–1294.
• D’Ambrosio, L. (2005). Hardy-type inequalities related to degenerate elliptic differential operators. Annali della Scuola Normale Superiore di Pisa, 4, 451–586.
• Folland, G. B. (1973). A fundamental solution for a subelliptic operator. Bulletin of the American Mathematical Society, 79, 373–376.
• Garcia A. J. & Peral A. I. (1998). Hardy inequalities and some critical elliptic and parabolic problems. Journal of Differential Equations, 144, 441–476.
• Goldstein J. A. & Kömbe I. (2003). Nonlinear parabolic differential equations with singular lower order term. Advances in Differential Equations, 10, 1153-1192.
• Goldstein, G. R., Goldstein, J. A. & Kombe, I. (2005). Nonlinear parabolic equations with singular coefficient and critical exponent. Applicable Analysis, 84(6), 571–583.
• Greiner, P. C. (1979). A fundamental solution for non-elliptic partial differential operator. Canadian Journal of Mathematics, 31, 1107–1120.
• Hörmander, L. (1967). Hypoelliptic second order differential equations. Acta Mathematica, 119, 147–171.
• Kombe, I., (2006). On the nonexistence of positive solutions to nonlinear degenerate parabolic equations with singular coefficients. Applicable Analysis, 85(5), 467-478.
• Lian, B. (2013). Some sharp Rellich type inequalities on nilpotent groups and application. Acta Mathematica Sinica, English Series, 33, 59–74.
• Nagel, S., Stein, E. M. & Wainger, S. (1985). Balls and metrics defined by vector fields I: Basic properties. Acta Mathematica, 155, 103–147.
• Niu, P., Ou, Y. & Han, J. (2010), Several Hardy-type inequalities with weights related to generalized Greiner operator. Canadian Mathematical Bulletin, 53, 153–162.
• Shen, Y. T. (1980). On the Dirichlet problem for quasilinear elliptic equation with strongly singular coefficients. Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, 1407-1417.
• Utku A. (2022). Greiner operatörü ile ilişkilendirilmiş hızlı difüzyon denklemi ve bazı integral eşitsizlikleri [Yüksek Lisans Tezi]. İstanbul Ticaret Üniversitesi Fen Bilimleri Enstitüsü, İstanbul.
• Yener, A. (2018). General weighted Hardy-type inequalities related to Greiner operators. Rocky Mountain Journal of Mathematics, 48(7), 2405-2430.
• Zhang, H. & Niu, P. (2003). Hardy-type inequalities and Pohozaev-type identities for a class of p-degenerate subelliptic operators and applications. Nonlinear Analysis, 54, 165–186.