Degenerate-singular elliptic operators: sharp coercivity, weighted well-posedness, and Green’s function methods for tumor-diffusion models

Eshmamatova, D.B., Ochilova, N.K., Эшмаматова, Д.Б., Очилова, Н.К.

Ҳисоблаш ва амалий математика муаммолари · 2026-yil

Annotatsiya

We study a family of degenerate-singular second- and third-order elliptic boundary value problems modeling anomalous diffusion in heterogeneous tumor tissue, all built on one differential operator with a power-law degeneracy in one spatial direction and a singular lower-order term in the other. First, in an anisotropically weighted Sobolev space matched to the degenerate direction, we prove existence and uniqueness of a weak solution of the nonlinear problem for small degeneracy exponents under explicit smallness conditions on the singular coefficient and on the growth of the monotone reaction term, together with interior regularity and a conditional global weighted estimate for larger exponents. Second, in an auxiliary weighted energy space adapted to the degenerate principal part, we establish a coercivity threshold for the singular coefficient that is valid for both signs of the coefficient and for every degeneracy exponent, and prove that its negative side is sharp by means of an explicit minimizing sequence for the underlying Hardy inequality; well-posedness, stability estimates, and interior regularity then follow. Third, for the associated third-order equation we formulate Dirichlet- and Neumanntype boundary value problems, prove a new extremum principle establishing uniqueness, and reduce existence, via the classical Carleman–Vekua regularization, to a Fredholm equation of the second kind built from an explicit Green’s function. A finite-difference scheme, verified against a manufactured exact solution with second-order convergence, confirms the behaviour of solutions near the degenerate boundary. The biomedical interpretation — diffusion of oxygen, a drug, or a signalling molecule through necrotic, poorly vascularized tumor tissue — is discussed throughout.

Maqola ma’lumotlari
MualliflarEshmamatova, D.B., Ochilova, N.K., Эшмаматова, Д.Б., Очилова, Н.К.
JurnalҲисоблаш ва амалий математика муаммолари
NashriyotТошкент ахборот технологиялари университети қошидаги, Ахборот-коммуникация технологиялари илмий-инновацион, маркази
ISSN2181-8460, 2181-046X
Nashr sanasi2026-09-15
Son4
Betlar133-151
TilIngliz
Ilmiy sohaFizika-matematika
DOI10.71310/pcam.4_74.2026.09
Indeksga qo‘shilgan2026-09-16

DOI: 10.71310/pcam.4_74.2026.09 · Maqolaning asl sahifasi

Kalit so‘zlar

degenerate elliptic equation, weighted Sobolev space, Muckenhoupt weight, Hardy inequality, Green’s function, Carleman–Vekua regularization, вырожденное эллиптическое уравнение, весовое пространство Соболева, вес Макенхаупта, неравенство Харди, функция Грина, регуляризация Карлемана — Векуа

Ilmiy soha

Fizika-matematika

Iqtibos keltirish

APAEshmamatova, D.B., Ochilova, N.K., Эшмаматова, Д.Б., & Очилова, Н.К. (2026). Degenerate-singular elliptic operators: sharp coercivity, weighted well-posedness, and Green’s function methods for tumor-diffusion models. Ҳисоблаш ва амалий математика муаммолари, (4), 133-151. https://doi.org/10.71310/pcam.4_74.2026.09
GOST (OAK)Eshmamatova, D.B. va b. Degenerate-singular elliptic operators: sharp coercivity, weighted well-posedness, and Green’s function methods for tumor-diffusion models // Ҳисоблаш ва амалий математика муаммолари. – 2026. – № 4. – P. 133-151. – DOI: 10.71310/pcam.4_74.2026.09.
MLAEshmamatova, D.B., Ochilova, N.K., Эшмаматова, Д.Б., and Очилова, Н.К. “Degenerate-singular elliptic operators: sharp coercivity, weighted well-posedness, and Green’s function methods for tumor-diffusion models.” Ҳисоблаш ва амалий математика муаммолари, (4), 133-151, 2026. https://doi.org/10.71310/pcam.4_74.2026.09
ChicagoEshmamatova, D.B., Ochilova, N.K., Эшмаматова, Д.Б., and Очилова, Н.К. “Degenerate-singular elliptic operators: sharp coercivity, weighted well-posedness, and Green’s function methods for tumor-diffusion models.” Ҳисоблаш ва амалий математика муаммолари, (4), 133-151 (2026). https://doi.org/10.71310/pcam.4_74.2026.09
HarvardEshmamatova, D.B., Ochilova, N.K., Эшмаматова, Д.Б., and Очилова, Н.К (2026) ‘Degenerate-singular elliptic operators: sharp coercivity, weighted well-posedness, and Green’s function methods for tumor-diffusion models’, Ҳисоблаш ва амалий математика муаммолари, (4), 133-151. https://doi.org/10.71310/pcam.4_74.2026.09
BibTeX
@article{eshmamatovadb2026,
  title = {Degenerate-singular elliptic operators: sharp coercivity, weighted well-posedness, and Green’s function methods for tumor-diffusion models},
  author = {Eshmamatova, D.B. and Ochilova, N.K. and Эшмаматова, Д.Б. and Очилова, Н.К.},
  journal = {Ҳисоблаш ва амалий математика муаммолари},
  year = {2026},
  number = {4},
  pages = {133-151},
  doi = {10.71310/pcam.4_74.2026.09},
  url = {https://doi.org/10.71310/pcam.4_74.2026.09}
}

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