Cazenave-Dickstein-Weissler-type extension of Fujita's problem on Heisenberg groups

2025/06/12 by Mokhtar Kirane, Kirane, Mokhtar, Ahmad Z. Fino +5 · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.10611

openalex publication_date 2025/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper examines the critical exponents for the existence of global solutions to the equation u_t-Δ_ℍu=∫₀^t(t-s)^-γ|u(s)|^p-1u(s) ds, · amp; 0≤γ · lt;1, η∈ ℍⁿ, t · gt;0, on the Heisenberg groups ℍⁿ. There exists a critical exponent p_c= max\\frac1γ,p_γ\∈(0,+∞], \hboxwith p_γ=1+(2(2-γ))/(Q-2+2γ), Q=2n+2 such that for all 1p_c, a global positive solution exists if the initial data is sufficiently small. The results obtained are a natural extension of the results of Cazenave et al. [Nonlinear Analysis 68 (2008), 862-874], where similar studies were carried out in ℝⁿ. Also given are several theorems concerning the lifespan estimates of local solutions for different cases of initial data. The proofs of the main results are based on test function methods and Banach fixed point principle.

Citations

• Fujita type results for a parabolic inequality with a non-linear convolution term on the Heisenberg group
• The Fujita exponent for a semilinear heat equation with forcing term on Heisenberg Group
• The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity
• Finite time blow up for wave equations with strong damping in an exterior domain
• The heat kernel and frequency localized functions on the Heisenberg group

Cited by

• Parabolic problems whose Fujita critical exponent is not given by scaling
• The Fujita exponent for a heat equation with mixed local and nonlocal nonlinearities on the Heisenberg group

Related

• The Fujita exponent for a semilinear heat equation with forcing term on Heisenberg Group
• The Fujita exponent for a heat equation with mixed local and nonlocal nonlinearities on the Heisenberg group
• Fujita type results for a parabolic inequality with a non-linear convolution term on the Heisenberg group
• The Existence of Global Solution for a Class of Semilinear Equations on Heisenberg Group
• Existence of solutions semilinear parabolic equations with singular initial data in the Heisenberg group