DEGENERATE PARABOLIC EQUATIONS ASSOCIATED WITH THE GREINER OPERATOR: NONEXISTENCE RESULTS VIA HARDY–LERAY INEQUALITIES
Evolution Equations and Control Theory, cilt.25, ss.309-330, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 25
- Basım Tarihi: 2026
- Doi Numarası: 10.3934/eect.2026092
- Dergi Adı: Evolution Equations and Control Theory
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Sayfa Sayıları: ss.309-330
- Anahtar Kelimeler: Critical exponents, Greiner operator, Hardy–Leray inequalities, nonexistence, positive solutions
- Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
- İstanbul Ticaret Üniversitesi Adresli: Evet
Özet
We study a nonlinear subelliptic parabolic equation associated with the Greiner operator ∆k, ut = ∆k(um) + V (w)um + η um|∇ku|q + λup in Ω × (0, T), subject to homogeneous Dirichlet boundary conditions on a Carnot–Carathéodory metric ball Ω. Here m, p, q > 0, η, λ ∈ R, and V ∈ L1loc(Ω) may exhibit singular behavior. Our analysis focuses on the nonexistence of positive local solutions. The criterion is formulated in terms of the bottom of the L2 spectrum of the operator −∆k − (1 − ε) V . In particular, we show that the absence of positive local solutions is closely related to the spectral properties of this operator. We also obtain a weighted Lp Hardy inequality with an optimal leading constant (p−p1)p and a positive remainder term. We further establish an Lp Hardy–Sobolev inequality with critical exponent p∗(s, p) = p(Q − s)/(Q − p) interpolating between the Sobolev embedding and the weighted Lp Hardy inequality. Finally, we prove a unified L2 Hardy–Leray inequality in which the constants (Q2−2)2 and 14 are simultaneously sharp. As applications of the nonexistence theorem, we analyze several classes of critical potentials arising from sharp functional inequalities. In particular, we determine explicit critical thresholds for Leray, boundary Hardy, Hardy, and perturbed Hardy potentials.