Systems of semilinear evolution inequalities with temporal fractional derivative on the Heisenberg group

Résumé

We investigate nonexistence results of nontrivial solutions of fractional differential inequalities of the form $$\bigl(\mathrm{FS}^{m}_{q}\bigr)\mbox{:}\quad \left \{ \textstyle\begin{array}{l} \mathbf{D}^{q}_{0/t}x_{i}-\Delta_{\mathbb{H}}(\lambda_{i}x_{i}) \geq {|\eta|}^{\alpha_{i+1}} {| x_{i+1} |}^{\beta_{i+1}}, \quad (\eta ,t) \in{\mathbb{H}}^{N}\times\, ]0,+\infty [ , 1 \leq i \leq m, \\ x_{m+1}=x_{1} , \end{array}\displaystyle \right . $$ where $\mathbf{D}^{q}_{0/t}$ is the time-fractional derivative of order $q \in(1,2)$ in the sense of Caputo, $\Delta_{\mathbb{H}}$ is the Laplacian in the $(2N+1)$ -dimensional Heisenberg group ${\mathbb {H}}^{N}$ , ${|\eta|}$ is the distance from η in ${\mathbb {H}}^{N}$ to the origin, $m\geq2$ , $\alpha_{m+1}=\alpha_{1}$ , $\beta _{m+1}=\beta_{1}$ , and $\lambda_{i}\in L^{\infty}({\mathbb{H}}^{N} \times\, ]0,+\infty [ )$ , $1 \leq i \leq m$ . The main results are concerned with $Q \equiv2N + 2$ , less than the critical exponents that depend on q, $\alpha_{i}$ , and $\beta_{i}$ , $1 \leq i \leq m$ . For $q=2$ , we deduce the results given by El Hamidi and Kirane (Abstr. Appl. Anal. 2004(2):155-164, 2004) and El Hamidi and Obeid (J. Math. Anal. Appl. 208(1):77-90, 2003) from the hyperbolic systems. For $m=1$ , we study the scalar case $$(\mathrm{FI}_{q})\mbox{:}\quad \mathbf{D}^{q}_{0/t}x - \Delta_{\mathbb{H}}(\lambda x) \geq {|\eta|}^{\alpha} {| x |}^{\beta}, $$ where $\beta>1$ , α are real parameters. In the last case, for $q=2$ , we return to the approach of Pohozaev and Véron (Manuscr. Math. 102:85-99, 2000) from the hyperbolic inequalities.

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