Zineb Sabbagh | Searcharxiv

arXiv2025

The Fujita exponent for a heat equation with mixed local and nonlocal nonlinearities on the Heisenberg group

This article deals with the problems of local and global solvability for a semilinear heat equation on the Heisenberg group involving a mixed local and nonlocal nonlinearity. The characteristic features of such equations, arising from the interplay between the geometric structure of the Heisenberg group and the combined nonlinearity, are analyzed in detail. The need to distinguish between subcritical and supercritical regimes is identified and justified through rigorous analysis. On the basis of the study, the author suggests precise conditions under which local-in-time mild solutions exist uniquely for regular, nonnegative initial data. It is proved that global existence holds under appropriate growth restrictions on the nonlinear terms. To complement these results, it is shown, by employing the capacity method, that solutions cannot exist globally in time when the nonlinearity exceeds a critical threshold. As a result, the Fujita exponent is formulated and identified as the dividing line between global existence and finite-time blow-up. In addition, lifespan estimates were obtained in the supercritical regime, providing insight into how the size of the initial data influences the time of blow-up.

math.AP↗

arXiv2025

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

This paper examines the critical exponents for the existence of global solutions to the equation \begin{equation*} \begin{array}{ll} \displaystyle u_t-\Delta_{\mathbb{H}}u=\int_0^t(t-s)^{-\gamma}|u(s)|^{p-1}u(s)\,ds,&\qquad 0\leq\gamma<1,\,\,\, {\eta\in \mathbb{H}^n,\,\,\,t>0,} \end{array}\end{equation*} on the Heisenberg groups $\mathbb{H}^n.$ There exists a critical exponent $$p_c= \max\Big\{\frac{1}{\gamma},p_\gamma\Big\}\in(0,+\infty],\quad\hbox{with}\quad p_\gamma=1+\frac{2(2-\gamma)}{Q-2+2\gamma},\,\,Q=2n+2$$ such that for all $1 p_c$, 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 $\mathbb{R}^n$. 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.

math.AP↗