Cazenave-Dickstein-Weissler-type extension of Fujita's problem on Heisenberg groups
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 $1<p\leq p_c,$ no global solution exists regardless of the non-negative initial data, while for $p>p_c$, 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.
Mokhtar Kirane、Ahmad Z. Fino、Berikbol T. Torebek、Zineb Sabbagh
数学
Mokhtar Kirane,Ahmad Z. Fino,Berikbol T. Torebek,Zineb Sabbagh.Cazenave-Dickstein-Weissler-type extension of Fujita's problem on Heisenberg groups[EB/OL].(2025-06-12)[2025-07-19].https://arxiv.org/abs/2506.10611.点此复制
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