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

详细信息    查看全文

• 作者：Bekkar Meneceur ; Kamel Haouam ; Amar Debbouche
• 关键词：critical exponent ; fractional derivative ; Heisenberg group ; evolution inequalities ; test function method
• 刊名：Advances in Difference Equations
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1455KB
• 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
• 出版者：Springer International Publishing
• ISSN：1687-1847
• 卷排序：2017

文摘

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.Keywordscritical exponentfractional derivativeHeisenberg groupevolution inequalitiestest function methodMSC35A0135B3335R0335R1135R451 IntroductionPohozaev and Véron [3] have established the question of nonexistence results for solutions of semilinear hyperbolic inequalities of the type $$ \frac{\partial^{2}x}{\partial t^{2}}-\Delta_{\mathbb{H}}(\lambda x)\geq| \eta|_{\mathbb{H}}^{\alpha}|x|^{\beta}, $$ (1) it is shown that no weak solution x exists provided that $$ \int_{\mathbb{R}^{2N+1}}x_{1}(\eta)\, d\eta\geq0 ,\qquad \alpha>-2 \quad \mbox{and} \quad 1< \beta\leq\frac{Q+1+\alpha}{Q-1 } $$ (2) In [1], El Hamidi and Kirane presented analogous results for a system of m hyperbolic semilinear inequalities of the form $$ \bigl(\mathrm{HS}^{m}\bigr)\mbox{:}\quad \left \{ \textstyle\begin{array}{l} \frac{\partial^{2}x_{i}}{\partial t^{2}} - \Delta_{\mathbb{H}}(\lambda _{i}x_{i}) \geq {|\eta|}^{\alpha_{i+1}} {| x_{i+1} | }^{\beta_{i+1}} , \\ (\eta,t) \in {\mathbb{H}}^{N}\times\, ]0,+\infty [,\quad 1 \leq i \leq m , \\ x_{m+1} = x_{1} , \end{array}\displaystyle \right . $$ (3) and expressed the Fujita exponent (see [4–6]), which ensures the system (\(\mathrm{HS}^{m}\)) admits no solution defined in \({\mathbb{H}}^{N}\) whenever \(Q \leq1+ \max (X_{1},X_{2},\ldots,X_{m} )\), where \((X_{1},X_{2},\ldots, X_{m})^{T}\) for the solution of the linear system (27).Their results have been generalized by El Hamidi and Obeid [2] to a system of m semilinear inequalities with higher-order time derivative of the type $$ \bigl(\mathrm{S}^{m}_{k}\bigr)\mbox{:}\quad \left \{ \textstyle\begin{array}{l} \frac{\partial^{k}x_{i}}{\partial t^{k}}-\Delta_{\mathbb{H}}(\lambda _{i}x_{i}) \geq {|\eta|}^{\alpha_{i+1}} {| x_{i+1} | }^{\beta_{i+1}} , \\ (\eta,t) \in {\mathbb{H}}^{N}\times\, ]0,+\infty [,\quad 1 \leq i \leq m , \\ x_{m+1}=x_{1},\quad k=1,2,\ldots, \end{array}\displaystyle \right . $$ (4) where they proved that the system (\(\mathrm{S}^{m}_{k}\)) admits no solution defined in \({\mathbb{H}}^{N}\) whenever \(Q \leq2 (1-\frac {1}{k} )+ \max (X_{1},X_{2},\ldots,X_{m} )\). Different works on the importance of inequalities can be found in [7, 8].In this paper, we generalize these results (for (\(\mathrm{HS}^{m}\))) to an evolution system with temporal fractional derivative of the form $$ \bigl(\mathrm{FS}^{m}_{q}\bigr)\mbox{:}\quad \left \{ \textstyle\begin{array}{l} \mathbf{D}^{q}_{0/t}x_{i}-\Delta_{\mathbb{H}}(\lambda_{i}x_{i}) \geq {|\eta|}^{\alpha_{i+1}} {| x_{i+1} |}^{\beta_{i+1}} , \\ (\eta,t) \in {\mathbb{H}}^{N}\times\, ]0,+\infty [, \quad 1 \leq i \leq m, \\ x_{m+1}=x_{1} q \in(1,2 ), \end{array}\displaystyle \right . $$ (5) and we show under certain initial conditions that the system (\(\mathrm{FS}^{m}_{q}\)) admits no solution defined in \({\mathbb{H}}^{N}\) whenever \(Q < Q^{\bullet}_{q}=2 (1-\frac{1}{q} )+ \max (X_{1},X_{2},\ldots,X_{m} )\).This paper is organized as follows. In Section 2, we present some essential facts from fractional calculus, more precisely, the definitions of the fractional derivative in the sense of Riemann-Liouville and in sense of Caputo and their relationship between them, for some new senses: the reader may refer to [9–11]. We also give some preliminaries as regards the Heisenberg group \(\mathbb {H}^{N}\) and the operator \(\Delta_{\mathbb{H}}\). In Section 3, we study the case of two inequalities. In Section 4, we study the general case of \(m>2\), and in the last Section 5, we study the scalar case.