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

Abstract

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.

1 Introduction

Pohozaev 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.

2 Notation and preliminaries

In this section, we present some known facts about the time-fractional derivative $\mathbf{D}^{q}_{0/t}$, the Heisenberg group ${\mathbb {H}}^{N}$ and the operator $\Delta_{\mathbb{H}}$.

The left-sided derivative and the right-sided derivative in the sense of Riemann-Liouville for $\psi\in L^{1}(0,T)$, of order $q \in(1,2)$ are defined, respectively, as follows:

$$\begin{aligned}& \bigl( D^{q}_{0/t}\psi \bigr) (t)=\frac{1}{\Gamma(2-q)} \biggl( \frac {d}{dt} \biggr)^{2} \int_{0}^{t}\frac{\psi(\sigma)}{(t-\sigma )^{q-1}}\,d\sigma, \\ & \bigl( D^{q}_{t/T}\psi \bigr) (t)=\frac{1}{\Gamma(2-q)} \biggl( \frac {d}{dt} \biggr)^{2} \int_{t}^{T}\frac{\psi(\sigma)}{(\sigma -t)^{q-1}}\,d\sigma, \end{aligned}$$

where Γ is the Euler gamma function.

If $\psi'' \in L^{1}(0,T)$, the derivative in the sense of Caputo of order $q \in(1,2)$ is defined by

$$\bigl( \mathbf{D}^{q}_{0/t}\psi \bigr) (t)=\frac{1}{\Gamma(2-q)} \int _{0}^{t}\frac{\psi''(\sigma)}{(t-\sigma)^{q-1}}\,d\sigma, $$

which is related to the Riemann-Liouville derivative by

$$\mathbf{D}^{q}_{0/t}\psi(t)=D^{q}_{0/t} \bigl(\psi(t)-\psi(0)-t\psi '(0) \bigr). $$

We also recall the formula of integration by parts if $0<\delta<1$:

$$\int_{0}^{T}\varphi(t) \bigl( D^{\delta}_{0/t} \psi \bigr) (t)\,dt= \int _{0}^{T} \bigl( D^{\delta}_{t/T} \varphi \bigr) (t)\psi(t)\,dt. $$

To derive the weak formulations, we have made use of the relations (see (2.30) and (2.31), p.37 in[12]):

$$\begin{aligned}& D^{1+q} _{o/t}\psi = D D^{q} _{o/t}\psi, \quad q \in(0,1), \end{aligned}$$

(6)

$$\begin{aligned}& D^{1+q} _{t/T}\psi = -D D^{q} _{t/T}\psi, \quad q \in(0,1), \end{aligned}$$

(7)

we also have the following formula (see Lemma 2.2, p.35 in [12]), for any $\delta\in(0,1)$:

$$ D^{\delta}_{t/T}\psi(t)=\frac{1}{\Gamma(1-\delta)} \biggl( \frac{\psi (T)}{(T-t)^{\delta}}- \int_{t}^{T}\frac{\psi'(\sigma)}{(\sigma-t)^{\delta }}\,d\sigma \biggr). $$

(8)

More details of fractional derivatives can be found in [5, 12, 13]; see also [14–16].

The Heisenberg group $\mathbb{H}^{n}$ of the dimension $(2N+1)$ is the space

$$\mathbb{R}^{2N+1}= \bigl\{ \eta=(x,y,\tau)\in\mathbb{R}^{N} \times \mathbb{R}^{N} \times\mathbb{R} \bigr\} $$

equipped with the group operation ‘∘’ defined by

$$ \eta\circ\tilde{\eta}= \Biggl(x+\tilde{x},y+\tilde{y},\tau +\tilde{\tau}+2 \sum _{i=1}^{N}(x_{i} \tilde{y}_{i}-\tilde {x}_{i}y_{i}) \Biggr), $$

(9)

where

$$\begin{aligned}& \eta=(x,y,\tau)= (x_{1},x_{2},\ldots,x_{N},y_{1},y_{2}, \ldots ,y_{N},\tau ), \\& \tilde{\eta} =(\tilde{x},\tilde{y},\tilde{\tau})= (\tilde{x}_{1}, \tilde{x}_{2},\ldots,\tilde{x}_{N},\tilde {y}_{1}, \tilde{y}_{2},\ldots,\tilde{y}_{N},\tilde{\tau } ), \end{aligned}$$

this group operation makes $\mathbb{H}^{n}$ have the structure of a Lie group.

The subelliptic Laplacian $\Delta_{\mathbb{H}}$ over $\mathbb{H}^{n}$ is defined by

$$ \Delta_{\mathbb{H}}=\sum_{i=1}^{N} \bigl(X_{i}^{2}+Y_{i}^{2} \bigr), $$

(10)

where

$$X_{i}=\frac{\partial}{\partial x_{i}}+2y_{i}\frac{\partial}{\partial \tau}\quad \mbox{and}\quad Y_{i}=\frac{\partial}{\partial y_{i}}-2x_{i} \frac {\partial}{\partial\tau}; $$

with a simple calculation, we can write

$$\Delta_{\mathbb{H}}=\sum_{i=1}^{N} \biggl(\frac{\partial^{2}}{\partial x_{i}^{2}}+\frac{\partial^{2}}{\partial y_{i}^{2}}+4y_{i} \frac{\partial ^{2}}{\partial x_{i}\, \partial\tau}-4x_{i}\frac{\partial^{2}}{\partial y_{i}\, \partial\tau}+4\bigl(x_{i}^{2}+y_{i}^{2} \bigr)\frac{\partial^{2}}{\partial \tau^{2}} \biggr). $$

The operator $\Delta_{\mathbb{H}}$ is a degenerate elliptic operator satisfying the Hörmander condition of order 1 (see [17]). It is invariant with respect to the left multiplication in the group since

$$\Delta_{\mathbb{H}} \bigl(x(\eta\circ\tilde{\eta}) \bigr)= ( \Delta_{\mathbb{H}}x ) (\eta\circ\tilde{\eta})\quad \forall(\eta, \tilde{\eta}) \in\mathbb{H}^{N} \times\mathbb{H}^{N}. $$

The distance between a point and the origin in $\mathbb{H}^{N}$ is defined by

$$|\eta|_{\mathbb{H}}= \Biggl(\tau^{2}+\sum _{i=1}^{N}\bigl(x_{i}^{2}+y_{i}^{2} \bigr)^{2} \Biggr)^{1/4}. $$

The application $\eta\rightarrow|\eta|_{\mathbb{H}}$ is homogeneous of degree one with respect to the natural group of dilatations

$$ \delta_{\lambda}(\eta)= \bigl(\lambda x,\lambda y, \lambda^{2} t \bigr). $$

(11)

We also know that the operator $\Delta_{\mathbb{H}}$ is homogeneous of degree 2 relative to the distance $\delta_{\lambda}$ given in (11), that is,

$$\Delta_{\mathbb{H}}=\lambda^{2}\delta_{\lambda} ( \Delta_{\mathbb {H}} ). $$

Obviously, the action of $\Delta_{\mathbb{H}}$ where the functions only depend on $\rho=|\eta|_{\mathbb{H}}$ is

$$\Delta_{\mathbb{H}}x(\rho)=a(\eta) \biggl(\frac{d^{2}x}{d\rho^{2}}+ \frac {(Q-1)}{\rho}\frac{dx}{d\rho} \biggr), $$

where

$$a(\eta)=\sum_{i=1}^{N}\frac{(x_{i}^{2}+y_{i}^{2})}{\rho^{2}} \quad \mbox{and}\quad Q=2N+2. $$

The number Q defined above is called the homogeneous dimension $\mathbb{H}^{N}$.

We also identify the points $\mathbb{H}^{N}$ with those of $\mathbb {R}^{2N+1}$, and we refer to the natural measurement of Hâar in $\mathbb{H}^{N}$ similar to that of Lebesgue $d\eta= dx \,dy \,d\tau$ in $\mathbb{R}^{2N+1}$. Readers can refer to [17–22] for more details of the analysis of the Heisenberg group.

3 Systems of two inequalities

In this section, we are interested with systems of type

$$ \bigl(\mathrm{FS}^{2}_{q}\bigr)\mbox{:}\quad \left \{ \textstyle\begin{array}{l} \mathbf{D}^{q}_{0/t}x - \Delta_{\mathbb{H}}(\lambda_{1}x) \geq |\eta |_{\mathbb{H}}^{\alpha_{1}}{| y |}^{\beta_{1}}\quad \mbox{in } \mathbb {H}^{n}\times\mathbb{R}^{+}, \\ \mathbf{D}^{q}_{0/t}y - \Delta_{\mathbb{H}}(\lambda_{2}y) \geq |\eta |_{\mathbb{H}}^{\alpha_{2}}{| x |}^{\beta_{2}} \quad \mbox{in } \mathbb {H}^{n}\times\mathbb{R}^{+}, \end{array}\displaystyle \right . $$

(12)

where $\mathbf{D}^{q}_{0/t}$ denotes the time-fractional derivative of order $q \in(1,2)$, in the sense of Caputo. The functions $\lambda_{1}$ and $\lambda_{2}$ introduced in (12) are assumed to be measurable and bounded functions on $\mathbb {H}^{n}\times\mathbb{R}^{+}$, where the exponents $\alpha_{1}$, $\alpha _{2}$ and $\beta_{1},\beta_{2}>1$ are real numbers. We denote by ${D}^{q}_{0/t}$, the time-fractional derivative of order $q \in(1,2)$ in the sense of Riemann-Liouville. The following holds.

Definition 3.1

Let $\lambda_{1}$ and $\lambda_{2}$ be two bounded measurable functions in $Q_{T}=\mathbb{R}^{2N+1}\times(0,T)$. A weak solution $(x,y)$ of the system ($\mathrm{FS}^{2}_{q}$) with positive initial data $x_{0},x_{1},y_{0},y_{1} \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{2N+1})$ is a pair of locally integrable functions $(x,y)$ such that $(x,y) \in L^{\beta _{2}}(Q_{T}, |\eta|_{\mathbb{H}}^{\alpha_{2}}\,d\eta \,dt)\times L^{\beta _{1}}(Q_{T},|\eta|_{\mathbb{H}}^{\alpha_{1}}\,d\eta \,dt)$ satisfying

$$ \left\{ \begin{aligned} & \int_{Q_{T}} \bigl(-xD^{q}_{t/T}\varphi+ \lambda_{1}x\Delta_{\mathbb {H}}\varphi+ |\eta|_{\mathbb{H}}^{\alpha_{1}}|y|^{\beta_{1}} \varphi +x_{1}(\eta)D^{q-1}_{t/T}\varphi \bigr)\,d \eta \,dt \\ &\quad {}+ \int_{\mathbb {R}^{2N+1}}x_{0}(\eta)D^{q-1}_{t/T} \varphi(0)\,d\eta\leq0, \\ & \int_{Q_{T}} \bigl(-yD^{q}_{t/T}\varphi+ \lambda_{2}y\Delta_{\mathbb {H}}\varphi+ |\eta|_{\mathbb{H}}^{\alpha_{2}}|x|^{\beta_{2}} \varphi +y_{1}(\eta)D^{q-1}_{t/T}\varphi \bigr)\,d \eta \,dt \\ &\quad {}+ \int_{\mathbb {R}^{2N+1}}y_{0}(\eta)D^{q-1}_{t/T} \varphi(0)\,d\eta\leq0 \end{aligned} \right . $$

(13)

for any nonnegative test function $\varphi\in C^{2}_{c}(Q_{T})$, such that $\varphi(\cdot,T)=D^{q -1}_{t/T}\varphi(\cdot,T)=0$.

Remark 3.2

We assume that the integrals in (13) are convergent. In Definition 3.1, if $T=+\infty$, then the solution is called global.

Theorem 3.3

Assume that

$$Q < Q^{\bullet}_{q}=2 \biggl(1-\frac{1}{q} \biggr)+ \frac{1}{\beta_{1}\beta _{2}-1}\max \bigl((\alpha_{1}+2)+\beta_{1}( \alpha_{2}+2),\beta_{2}(\alpha _{1}+2)+( \alpha_{2}+2) \bigr). $$

Then there is no weak nontrivial solution $(x,y)$ of the system ($\mathrm{FS}^{2}_{q}$).

Proof

By contradiction, we suppose $(x,y)$ to be a nontrivial weak solution of ($\mathrm{FS}^{2}_{q}$), which generally exists in time, that is, $(x,y)$ exists in $(0,T^{*})$ for an arbitrary $T^{*}$.

Let T and R be two positive real numbers such that $0< TR< T^{*}$.

Since the initial data $x_{0}$, $x_{1}$, $y_{0}$, $y_{1}$ are nonnegative, and $D^{q -1}_{t/T}\varphi\geq0$ (from (8)), the variational formulation (13) implies

$$ \left\{ \begin{aligned} & \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{1}}|y|^{\beta _{1}} \varphi \,d\eta \,dt \leq \int_{Q_{{TR}}} xD^{q}_{t/TR}\varphi \,d\eta \,dt - \int_{Q_{{TR}}} \lambda_{1}x\Delta_{\mathbb{H}}\varphi \,d\eta \,dt, \\ & \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{2}}|x|^{\beta _{2}} \varphi \,d\eta \,dt \leq \int_{Q_{{TR}}} yD^{q}_{t/TR}\varphi \,d\eta \,dt - \int_{Q_{{TR}}} \lambda_{2}y\Delta_{\mathbb{H}}\varphi \,d\eta \,dt. \end{aligned} \right . $$

From the Hölder inequality, we get

$$ \left\{ \begin{aligned} & \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{1}}|y|^{\beta _{1}} \varphi \,d\eta \,dt \\ &\quad \leq \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb {H}}^{\alpha_{2}}|x|^{\beta_{2}} \varphi \,d\eta \,dt \biggr)^{\frac{1}{\beta _{2}}} \biggl( \int_{Q_{{TR}}}\bigl\vert D^{q}_{t/TR}\varphi \bigr\vert ^{\beta _{2}'} \bigl( |\eta|_{\mathbb{H}}^{\alpha_{2}}\varphi \bigr)^{-\frac {\beta_{2}'}{\beta_{2}}}\,d\eta \,dt \biggr)^{\frac{1}{\beta_{2}'}} \\ &\qquad {}+\Vert \lambda_{1} \Vert _{\infty} \biggl( \int_{Q_{{TR}}} |\eta |_{\mathbb{H}}^{\alpha_{2}}|x|^{\beta_{2}} \varphi \,d\eta \,dt \biggr)^{\frac {1}{\beta_{2}}} \biggl( \int_{Q_{{TR}}}\vert \Delta_{\mathbb{H}}\varphi \vert ^{\beta_{2}'} \bigl( |\eta|_{\mathbb{H}}^{\alpha_{2}}\varphi \bigr)^{-\frac{\beta_{2}'}{\beta_{2}}}\,d\eta \,dt \biggr)^{\frac{1}{\beta _{2}'}} \end{aligned} \right . $$

and

$$ \left\{ \begin{aligned} & \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{2}}|x|^{\beta _{2}} \varphi \,d\eta \,dt \\ &\quad \leq \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb {H}}^{\alpha_{1}}|y|^{\beta_{1}} \varphi \,d\eta \,dt \biggr)^{\frac{1}{\beta _{1}}} \biggl( \int_{Q_{{TR}}}\bigl\vert D^{q}_{t/TR}\varphi \bigr\vert ^{\beta _{1}'} \bigl( |\eta|_{\mathbb{H}}^{\alpha_{1}}\varphi \bigr)^{-\frac {\beta_{1}'}{\beta_{1}}}\,d\eta \,dt \biggr)^{\frac{1}{\beta_{1}'}} \\ &\qquad {}+\Vert \lambda_{2} \Vert _{\infty} \biggl( \int_{Q_{{TR}}} |\eta |_{\mathbb{H}}^{\alpha_{1}}|y|^{\beta_{1}} \varphi \,d\eta \,dt \biggr)^{\frac {1}{\beta_{1}}} \biggl( \int_{Q_{{TR}}}\vert \Delta_{\mathbb{H}}\varphi \vert ^{\beta_{1}'} \bigl( |\eta|_{\mathbb{H}}^{\alpha_{1}}\varphi \bigr)^{-\frac{\beta_{1}'}{\beta_{1}}}\,d\eta \,dt \biggr)^{\frac{1}{\beta _{1}'}}. \end{aligned} \right . $$

Next, C denotes a constant which may vary from line to line but is independent on the terms which will take part in any limit process. So, we obtain

$$ \int_{Q_{{TR}}}|\eta|_{\mathbb{H}}^{\alpha_{1}}|y|^{\beta_{1}} \varphi \,d\eta \,dt \leq C \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha _{2}}|x|^{\beta_{2}} \varphi \,d\eta \,dt \biggr)^{\frac{1}{\beta_{2}}} \mathcal{A} $$

(14)

and

$$ \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{2}}|x|^{\beta _{2}} \varphi \,d\eta \,dt \leq C \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb {H}}^{\alpha_{1}}|y|^{\beta_{1}} \varphi \,d\eta \,dt \biggr)^{\frac{1}{\beta _{1}}} \mathcal{B}, $$

(15)

where

$$\begin{aligned}& \mathcal{A} = \biggl( \int_{Q_{{TR}}}\bigl\vert D^{q}_{t/TR}\varphi \bigr\vert ^{\beta_{2}'}\bigl( \vert \eta \vert _{\mathbb{H}}^{\alpha_{2}} \varphi \bigr)^{-\frac{\beta_{2}'}{\beta_{2}}}\,d\eta \,dt\biggr)^{\frac{1}{\beta_{2}'}} + \biggl( \int_{Q_{{TR}}}\vert \Delta_{\mathbb{H}}\varphi \vert ^{\beta _{2}'}\bigl( \vert \eta \vert _{\mathbb{H}}^{\alpha_{2}}\varphi \bigr)^{-\frac {\beta_{2}'}{\beta_{2}}}\,d\eta \,dt\biggr)^{\frac{1}{\beta_{2}'}}, \\& \mathcal{B} = \biggl( \int_{Q_{{TR}}}\bigl\vert D^{q}_{t/TR}\varphi \bigr\vert ^{\beta_{1}'}\bigl( \vert \eta \vert _{\mathbb{H}}^{\alpha_{1}} \varphi \bigr)^{-\frac{\beta_{1}'}{\beta_{1}}}\,d\eta \,dt\biggr)^{\frac{1}{\beta_{1}'}} + \biggl( \int_{Q_{{TR}}}\vert \Delta_{\mathbb{H}}\varphi \vert ^{\beta _{1}'}\bigl( \vert \eta \vert _{\mathbb{H}}^{\alpha_{1}}\varphi \bigr)^{-\frac {\beta_{1}'}{\beta_{1}}}\,d\eta \,dt\biggr)^{\frac{1}{\beta_{1}'}}; \end{aligned}$$

from (14), (15), we have

$$\begin{aligned}& \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{1}}|y|^{\beta _{1}} \varphi \,d\eta \,dt \biggr)^{1-\frac{1}{\beta_{1}\beta_{2}}} \leq C\mathcal{B}^{\frac{1}{\beta_{2}}} \mathcal{A}, \end{aligned}$$

(16)

$$\begin{aligned}& \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{2}}|x|^{\beta _{2}} \varphi \,d\eta \,dt \biggr)^{1-\frac{1}{\beta_{1}\beta_{2}}} \leq C \mathcal{A}^{\frac{1}{\beta_{1}}} \mathcal{B}. \end{aligned}$$

(17)

Now, we take

$$ \varphi(\eta,t)= \varphi(x,y,\tau,t)=\Phi \biggl( \frac{\tau^{2\theta }+|x|^{4\theta}+|y|^{4\theta}+t^{4}}{R^{4}} \biggr), $$

(18)

where $\Phi\in\mathcal{D}(\mathbb{R}^{+})$ is a smooth nonnegative test function which satisfies $0\leq\Phi\leq1$ and

$$ \Phi(r)= \left \{ \textstyle\begin{array}{l@{\quad}l} 0& \mbox{if } r\geq2, \\ 1& \mbox{if } 0\leq r \leq1. \end{array}\displaystyle \right . $$

(19)

Then $\theta>1$, which will be specified later.

Then

$$ \left\{ \begin{aligned} \Delta_{\mathbb{H}}\varphi(\eta,t)={}& \frac{4\theta\Phi'(\rho )}{R^{4}} \bigl[ \bigl(N+2(2\theta-1) \bigr) \bigl(|x|^{2(2\theta-1)} +|y|^{2(2\theta-1)} \bigr) \\ &{}+2(2\theta-1)\tau^{2(\theta-1)} \bigl(|x|^{2}+|y|^{2} \bigr) \bigr] \\ &{}+ \frac{16\theta^{2}\Phi''(\rho)}{R^{8}} \bigl[|x|^{2(4\theta -1)}+|y|^{2(4\theta-1)} +2 \tau^{2\theta-1}\langle x,y \rangle \bigl(|x|^{2(2\theta-1)}-|y|^{2(2\theta-1)} \bigr) \\ &{}+ \tau^{2(2\theta-1)} \bigl(|x|^{2}+|y|^{2} \bigr) \bigr], \end{aligned} \right . $$

where

$$\rho=\frac{\tau^{2\theta}+|x|^{4\theta}+|y|^{4\theta}+t^{4}}{R^{4}} $$

to estimate $\mathcal{A}$, $\mathcal{B}$ (in (16) and (17)), by changing variables: $(\eta,t)=(x,y,\tau,t)\longmapsto(\tilde{\eta} ,\tilde {t})=(\tilde{x},\tilde{y},\tilde{\tau},\tilde{t})$ where

$$ \tilde{x}=R^{-\frac{1}{\theta}}x,\qquad \tilde{y}=R^{-\frac{1}{\theta }}y, \qquad \tilde{\tau}=R^{-\frac{2}{\theta}}\tau,\qquad \tilde{t}=R^{-1}t. $$

(20)

We choose

$$ \Omega= \bigl\{ (\tilde{\eta} ,\tilde{t})=(\tilde {x},\tilde{y},\tilde{\tau}, \tilde{t}) \in\mathbb {H}^{N}\times\mathbb{R}^{+} :\tilde{ \tau}^{2}+|\tilde {x}|^{4}+|\tilde{y}|^{4}+ \tilde{t}^{\theta} < 2 \bigr\} . $$

Therefore,

$$ \bigl\vert \Delta_{\mathbb{H}}\varphi(\tilde{\eta},\tilde{t}) \bigr\vert \leq\frac{C}{R^{\frac{2}{\theta}}} \quad \forall(\tilde{\eta },\tilde{t}) \in \Omega. $$

(21)

As $d\eta \,dt= R^{\frac{2N+2}{\theta}+1}\,d\tilde{\eta}\,d\tilde{t}$ and $|\eta|_{\mathbb{H}}=R^{\frac{1}{\theta}} |\tilde{\eta }|_{\mathbb{H}}$, we establish the following estimates:

$$\begin{aligned}& \int_{Q_{{TR}}}\bigl\vert D^{q}_{t/TR}\varphi \bigr\vert ^{\beta_{2}'} \bigl( |\eta|_{\mathbb{H}}^{\alpha_{2}}\varphi \bigr)^{-\frac{\beta _{2}'}{\beta_{2}}}\,d\eta \,dt \\& \quad =R^{-q \beta_{2}'- \frac{\alpha_{2} \beta _{2}'}{\theta\beta_{2}}+\frac{2N+2}{\theta}+1} \int_{\Omega}\bigl\vert D^{q}_{\tilde{t}/T}\Phi\circ \tilde{\rho} \bigr\vert ^{\beta_{2}'} \bigl( |\tilde{\eta}|_{\mathbb{H}}^{\alpha _{2}} \Phi\circ\tilde{\rho} \bigr)^{-\frac{\beta_{2}'}{\beta _{2}}}\,d\tilde{\eta} \,d\tilde{t} \end{aligned}$$

(22)

and

$$\begin{aligned}& \int_{Q_{{TR}}}\vert \Delta_{\mathbb{H}}\varphi \vert ^{\beta _{2}'} \bigl( |\eta|_{\mathbb{H}}^{\alpha_{2}}\varphi \bigr)^{-\frac {\beta_{2}'}{\beta_{2}}}\,d\eta \,dt \\& \quad \leq C R^{-\frac{2}{\theta}\beta_{2}'- \frac{\alpha_{2}\beta_{2}'}{\theta\beta_{2}}+\frac{2N+2}{\theta}+1} \int_{\Omega} \bigl( |\tilde{\eta}|_{\mathbb{H}}^{\alpha_{2}} \Phi \circ\tilde{\rho} \bigr)^{-\frac{\beta_{2}'}{\beta _{2}}}\,d\tilde{\eta} \,d\tilde{t}. \end{aligned}$$

(23)

We choose θ as the right-hand side of (22) and (23) which are of the same order in R. For this purpose, we take $\theta=\frac {2}{q}$, therefore

$$\mathcal{A}\leq C R^{-q-\frac{q \alpha_{2}}{2\beta_{2}}+\frac{q}{2}\frac {2N+2}{\beta_{2}'} +\frac{1}{\beta_{2}'}}. $$

Similarly, we can get

$$\mathcal{B}\leq C R^{-q-\frac{q \alpha_{1}}{2\beta_{1}}+\frac{q}{2}\frac {2N+2}{\beta_{1}'} +\frac{1}{\beta_{1}'}}. $$

From (16) and (17), it follows that

$$\begin{aligned}& \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{1}}|y|^{\beta _{1}} \varphi \,d\eta \,dt \biggr)^{1-\frac{1}{\beta_{1}\beta_{2}}} \leq C R^{-q-\frac{q \alpha_{2}}{2\beta_{2}}+\frac{q}{2}\frac{2N+2}{\beta _{2}'} +\frac{1}{\beta_{2}'}+\frac{1}{\beta_{2}} [-q -\frac{q \alpha _{1}}{2\beta_{1}}+\frac{q}{2}\frac{2N+2}{\beta_{1}'} +\frac{1}{\beta _{1}'} ]}, \\& \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{2}}|x|^{\beta _{2}} \varphi \,d\eta \,dt \biggr)^{1-\frac{1}{\beta_{1}\beta_{2}}} \leq C R^{-q-\frac{q \alpha_{1}}{2\beta_{1}}+\frac{q}{2}\frac{2N+2}{\beta _{1}'} +\frac{1}{\beta_{1}'}+\frac{1}{\beta_{1}} [-q-\frac{q \alpha _{2}}{2\beta_{2}}+\frac{q}{2}\frac{2N+2}{\beta_{2}'} +\frac{1}{\beta _{2}'} ]}. \end{aligned}$$

Thus, we have

$$ \left\{ \begin{aligned} &{-}q-\frac{q \alpha_{2}}{2\beta_{2}}+ \frac{q}{2}\frac{2N+2}{\beta_{2}'} +\frac{1}{\beta_{2}'}+\frac{1}{\beta_{2}} \biggl[-q -\frac{q \alpha _{1}}{2\beta_{1}}+\frac{q}{2}\frac{2N+2}{\beta_{1}'} + \frac{1}{\beta _{1}'} \biggr] < 0,\quad \mbox{or} \\ &{-}q-\frac{q \alpha_{1}}{2\beta_{1}}+\frac{q}{2}\frac{2N+2}{\beta_{1}'} + \frac{1}{\beta_{1}'}+\frac{1}{\beta_{1}} \biggl[-q-\frac{q \alpha _{2}}{2\beta_{2}}+ \frac{q}{2}\frac{2N+2}{\beta_{2}'} +\frac{1}{\beta _{2}'} \biggr] < 0. \end{aligned} \right . $$

(24)

This condition is equivalent to

$$Q < Q^{\bullet}_{q}=2 \biggl(1-\frac{1}{q} \biggr)+ \frac{1}{\beta_{1}\beta _{2}-1}\max \bigl((\alpha_{1}+2)+\beta_{1}( \alpha_{2}+2),\beta_{2}(\alpha _{1}+2)+( \alpha_{2}+2) \bigr). $$

Finally, let $R\rightarrow\infty$, taking into account the estimations (14), (17) or (15), (16) and using the Fatou lemma, we get

$$\begin{aligned}& \int_{\mathbb{R}^{2N+1}} \int_{\mathbb{R}^{+}} |\eta|_{\mathbb {H}}^{\beta}|x|^{\beta} \,d\eta \,dt \leq 0, \end{aligned}$$

(25)

$$\begin{aligned}& \int_{\mathbb{R}^{2N+1}} \int_{\mathbb{R}^{+}} |\eta|_{\mathbb {H}}^{\beta}|y|^{\beta} \,d\eta \,dt \leq 0 . \end{aligned}$$

(26)

Therefore, $x\equiv0$ and $y\equiv0$, which is a contradiction. □

Corollary 3.4

Assume that

$$Q < Q^{\bullet}_{q}=2 \biggl(1-\frac{1}{q} \biggr)+ \max (X_{1},X_{2} ), $$

where the vector $(X_{1},X_{2})^{T}$ is the solution of the linear system

$$\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} -1 & \beta_{1} \\ \beta_{2} & -1 \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} X_{1} \\ X_{2} \end{array}\displaystyle \right ) =\left ( \textstyle\begin{array}{@{}c@{}} \alpha_{1}+2 \\ \alpha_{2}+2 \end{array}\displaystyle \right ). $$

Then there is no weak nontrivial solution $(x,y)$ of the system ($\mathrm{FS}^{2}_{q}$).

Proof

To get our result, we use the fact that the vector $(X_{1},X_{2})^{T}$ is given by

$$\left ( \textstyle\begin{array}{@{}c@{}} X_{1} \\ X_{2} \end{array}\displaystyle \right ) =\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} -1 & \beta_{1} \\ \beta_{2} & -1 \end{array}\displaystyle \right )^{-1} \left ( \textstyle\begin{array}{@{}c@{}} \alpha_{1}+2 \\ \alpha_{2}+2 \end{array}\displaystyle \right )=\frac{1}{\beta_{1}\beta_{2}-1} \left ( \textstyle\begin{array}{@{}c@{}} (\alpha_{1}+2)+\beta_{1}(\alpha_{2}+2) \\ \beta_{2}(\alpha_{1}+2)+(\alpha_{2}+2) \end{array}\displaystyle \right ). $$

 □

4 Systems of m inequalities

Let $(X_{1},X_{2},\ldots,X_{m})^{T}$ be the solution of the linear system

$$ \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} -1 & \beta_{1} & 0 & \ldots& 0 \\ 0 & -1 & \beta_{2} & \ddots& \vdots\\ \vdots& \ddots& \ddots& \ddots& 0 \\ 0 & 0 & \ddots& \ddots& \beta_{m-1} \\ \beta_{m} & 0 & \ldots& 0 & -1 \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} X_{1} \\ X_{2} \\ \vdots\\ X_{m-1} \\ X_{m} \end{array}\displaystyle \right )= \left ( \textstyle\begin{array}{@{}c@{}} \alpha_{1}+2 \\ \alpha_{2}+2 \\ \vdots\\ \alpha_{m-1}+2 \\ \alpha_{m}+2 \end{array}\displaystyle \right ), $$

(27)

where $\alpha_{i}$ and $\beta_{i} > 1$ are given real numbers, $i \in \{1, 2,\ldots,m \}$.

Consider the system

$$\bigl(\mathrm{FS}^{m}_{q}\bigr)\mbox{:}\quad \left \{ \textstyle\begin{array}{l} \mathbf{D}^{q}_{0/t}x_{i} - \Delta_{\mathbb{H}}(\lambda_{i}x_{i}) \geq {|\eta|}^{\alpha_{i+1}} {| x_{i+1} |}^{\beta_{i+1}} , \\ (\eta,t) \in {\mathbb{H}}^{N}\times\, ]0,+\infty [,\quad 1 \leq i \leq m, \\ x_{m+1} = x_{1}, \end{array}\displaystyle \right . $$

where $\beta_{m+1} = \beta_{1}$, $\alpha_{m+1} = \alpha_{1}$, and the initial data are

$$\left \{ \textstyle\begin{array}{l} x_{i}(\eta,0) = x^{(0)}_{i},\quad 1 \leq i \leq m, \\ \frac{\partial x_{i}}{\partial t}(\eta,0 ) = x^{(1)}_{i},\quad 1 \leq i \leq m. \end{array}\displaystyle \right . $$

Definition 4.1

Let $\lambda_{i}$, $i \in \{1, 2,\ldots,m \}$ be m bounded measurable functions in $Q_{T}=\mathbb{R}^{2N+1}\times(0,T)$. A weak solution $(x_{1},\ldots,x_{m})$ of the system ($\mathrm{FS}^{m}_{q}$) with positive initial data $(x^{(0)}_{i},x^{(1)}_{i} ) \in (L^{1}_{\mathrm{loc}}(\mathbb{R}^{2N+1}) )^{2}$, $i \in \{1, 2,\ldots ,m \}$, is a vector of locally integrable functions $(x_{1},\ldots,x_{m})$ such that $x_{i} \in L^{\beta_{i}}(Q_{T},|\eta|_{\mathbb{H}}^{\alpha _{i}}\,d\eta \,dt)$, $i \in \{1, 2,\ldots,m \}$, satisfying

$$ \left\{ \begin{aligned} & \int_{Q_{T}} \bigl(-x_{i}D^{q}_{t/T} \varphi+ \lambda_{i}x\Delta _{\mathbb{H}}\varphi+ | \eta|_{\mathbb{H}}^{\alpha _{i+1}}|x_{i+1}|^{\beta_{i+1}} \varphi+x^{(1)}_{i}(\eta )D^{q-1}_{t/T} \varphi \bigr)\,d\eta \,dt \\ &\quad {}+ \int_{\mathbb {R}^{2N+1}}x^{(0)}_{i}( \eta)D^{q-1}_{t/T}\varphi(0)\,d\eta\leq0, \quad i \in \{1, 2, \ldots,m-1 \} , \end{aligned} \right . $$

(28)

and

$$ \left\{ \begin{aligned} & \int_{Q_{T}} \bigl(-x_{m}D^{q}_{t/T} \varphi+ \lambda_{m}x\Delta _{\mathbb{H}}\varphi+ | \eta|_{\mathbb{H}}^{\alpha_{1}}|x_{1}|^{\beta _{1}} \varphi+x^{(1)}_{m}(\eta)D^{q-1}_{t/T} \varphi \bigr)\,d\eta \,dt \\ &\quad {}+ \int_{\mathbb{R}^{2N+1}}x^{(0)}_{m}(\eta)D^{q-1}_{t/T} \varphi(0)\,d\eta \leq0 \end{aligned} \right . $$

(29)

for any nonnegative test function $\varphi\in C^{2}_{c}(Q_{T})$, such that $\varphi(\cdot,T)=D^{q -1}_{t/T}\varphi(\cdot,T)=0$.

Theorem 4.2

If the following hypothesis holds:

$$Q < Q^{\bullet}_{q}=2 \biggl(1-\frac{1}{q} \biggr)+ \max (X_{1},X_{2},\ldots,X_{m} ), $$

then the system ($\mathrm{FS}^{m}_{q}$) does not have any weak nontrivial solution.

Proof

The proof is to be reduced to the case $m = 3$, the general case can be extended similarly.

Let $(x_{1},x_{2},x_{3})$ be a nontrivial weak solution of ($\mathrm{FS}^{3}_{q}$), as explained in the proof of Theorem 3.3, from the positivity of initial data and $D^{q-1}_{t/T}\varphi\geq0$, inequalities (28) and (29) imply that

$$ \left\{ \begin{aligned} & \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{1}}|x_{1}|^{\beta _{1}} \varphi \,d\eta \,dt \leq \int_{Q_{{TR}}} x_{3}D^{q}_{t/TR} \varphi \,d\eta \,dt - \int_{Q_{{TR}}} \lambda_{3}x_{3} \Delta_{\mathbb{H}}\varphi \,d\eta \,dt, \\ & \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{2}}|x_{2}|^{\beta _{2}} \varphi \,d\eta \,dt \leq \int_{Q_{{TR}}} x_{1}D^{q}_{t/TR} \varphi \,d\eta \,dt - \int_{Q_{{TR}}} \lambda_{1}x_{1} \Delta_{\mathbb{H}}\varphi \,d\eta \,dt, \\ & \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{3}}|x_{3}|^{\beta _{3}} \varphi \,d\eta \,dt \leq \int_{Q_{{TR}}} x_{2}D^{q}_{t/TR} \varphi \,d\eta \,dt - \int_{Q_{{TR}}} \lambda_{2}x_{2} \Delta_{\mathbb{H}}\varphi \,d\eta \,dt. \end{aligned} \right . $$

According to Hölder’s inequality, we obtain

$$\begin{aligned}& \int_{Q_{{TR}}}|\eta|_{\mathbb{H}}^{\alpha_{1}}|x_{1}|^{\beta _{1}} \varphi \,d\eta \,dt \leq C \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb {H}}^{\alpha_{3}}|x_{3}|^{\beta_{3}} \varphi \,d\eta \,dt \biggr)^{\frac {1}{\beta_{3}}} \mathcal{A}_{3}, \end{aligned}$$

(30)

$$\begin{aligned}& \int_{Q_{{TR}}}|\eta|_{\mathbb{H}}^{\alpha_{2}}|x_{2}|^{\beta _{2}} \varphi \,d\eta \,dt \leq C \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb {H}}^{\alpha_{1}}|x_{1}|^{\beta_{1}} \varphi \,d\eta \,dt \biggr)^{\frac {1}{\beta_{1}}} \mathcal{A}_{1}, \end{aligned}$$

(31)

and

$$ \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{3}}|x_{3}|^{\beta _{3}} \varphi \,d\eta \,dt \leq C \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb {H}}^{\alpha_{2}}|x_{2}|^{\beta_{2}} \varphi \,d\eta \,dt \biggr)^{\frac {1}{\beta_{2}}} \mathcal{A}_{2}, $$

(32)

where

$$\begin{aligned} \mathcal{A}_{i} =& \biggl( \int_{Q_{{TR}}}\bigl\vert D^{q}_{t/TR}\varphi \bigr\vert ^{\beta_{i}'} \bigl( |\eta|_{\mathbb{H}}^{\alpha_{i}}\varphi \bigr)^{-\frac{\beta_{i}'}{\beta_{i}}}\,d\eta \,dt \biggr)^{\frac{1}{\beta _{i}'}} \\ &{}+ \biggl( \int_{Q_{{TR}}}\vert \Delta_{\mathbb{H}}\varphi \vert ^{\beta _{i}'} \bigl( |\eta|_{\mathbb{H}}^{\alpha_{i}}\varphi \bigr)^{-\frac {\beta_{i}'}{\beta_{i}}}\,d\eta \,dt \biggr)^{\frac{1}{\beta_{i}'}},\quad i=1,2,3. \end{aligned}$$

From (30), (31), and (32), we get

$$\begin{aligned}& \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{1}}|x_{1}|^{\beta _{1}} \varphi \,d\eta \,dt \biggr)^{1-\frac{1}{\beta_{1}\beta_{2}\beta_{3}}} \leq C\mathcal{A}^{\frac{1}{\beta_{2}\beta_{3}}}_{1} \mathcal{A}^{\frac {1}{\beta_{3}}}_{2}\mathcal{A}_{3}, \end{aligned}$$

(33)

$$\begin{aligned}& \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{2}}|x_{2}|^{\beta _{2}} \varphi \,d\eta \,dt \biggr)^{1-\frac{1}{\beta_{1}\beta_{2}\beta_{3}}} \leq C\mathcal{A}^{\frac{1}{\beta_{1}\beta_{3}}}_{2} \mathcal{A}^{\frac {1}{\beta_{1}}}_{3}\mathcal{A}_{1}, \end{aligned}$$

(34)

$$\begin{aligned}& \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{3}}|x_{3}|^{\beta _{3}} \varphi \,d\eta \,dt \biggr)^{1-\frac{1}{\beta_{1}\beta_{2}\beta_{3}}} \leq C \mathcal{A}^{\frac{1}{\beta_{1}\beta_{2}}}_{3} \mathcal {A}^{\frac{1}{\beta_{2}}}_{1}\mathcal{A}_{2}. \end{aligned}$$

(35)

Applying the test function φ (18), and changing of variables (20), given in the proof of Theorem 3.3, we obtain

$$ \mathcal{A}_{i}\leq C R^{\sigma_{i}},\quad i=1,2,3, $$

such that

$$ \sigma_{i}=-q-\frac{q \alpha_{i}}{2 \beta_{i}}+\frac{q}{2 \beta_{i}'} Q + \frac{1}{\beta_{i}'},\quad i=1,2,3. $$

Therefore, from (33), (34), and (35), we get

$$\begin{aligned}& \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{1}}|x_{1}|^{\beta _{1}} \varphi \,d\eta \,dt \biggr)^{1-\frac{1}{\beta_{1}\beta_{2}\beta_{3}}} \leq C R^{\sigma_{3}+\frac{\sigma_{2}}{\beta_{3}}+\frac{\sigma _{1}}{\beta_{2}\beta_{3}}}, \end{aligned}$$

(36)

$$\begin{aligned}& \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{2}}|x_{2}|^{\beta _{2}} \varphi \,d\eta \,dt \biggr)^{1-\frac{1}{\beta_{1}\beta_{2}\beta_{3}}} \leq C R^{\sigma_{1}+\frac{\sigma_{3}}{\beta_{1}}+\frac{\sigma _{2}}{\beta_{1}\beta_{3}}} , \end{aligned}$$

(37)

$$\begin{aligned}& \biggl( \int_{Q_{{TR}}} |\eta|_{\mathbb{H}}^{\alpha_{3}}|x_{3}|^{\beta _{3}} \varphi \,d\eta \,dt \biggr)^{1-\frac{1}{\beta_{1}\beta_{2}\beta_{3}}} \leq C R^{\sigma_{2}+\frac{\sigma_{1}}{\beta_{2}}+\frac{\sigma _{3}}{\beta_{1}\beta_{2}}} . \end{aligned}$$

(38)

To end, the exponents of R in (36), (37), and (38) are strictly less than zero if and only if $Q<2(1- 1/q)+ \max(X_{1},X_{2},X_{3})$, where the vector $(X_{1},X_{2},X_{3})^{T}$ is the solution of

$$ \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} -1 & \beta_{1} & 0 \\ 0 & -1 & \beta_{2} \\ \beta_{3} & 0 & -1 \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} X_{1} \\ X_{2} \\ X_{3} \end{array}\displaystyle \right )= \left ( \textstyle\begin{array}{@{}c@{}} \alpha_{1} +2 \\ \alpha_{2}+2 \\ \alpha_{3} +2 \end{array}\displaystyle \right ). $$

(39)

We conclude that $(x_{1},x_{2},x_{3})\equiv(0, 0, 0)$. This contradicts the assertion. □

5 The scalar case

Let us consider the inequality of the form

$$ (\mathrm{FI}_{q})\mbox{:}\quad \left \{ \textstyle\begin{array}{l} \mathbf{D}^{q}_{0/t}(x) -\Delta_{\mathbb{H}}(\lambda x) \geq|\eta |_{\mathbb{H}}^{\alpha}|x|^{\beta} \quad \mbox{for } (\eta,t)\in\mathbb {H}^{N}\times\mathbb{R}, \\ x(\eta,0) = x_{0}(\eta)\geq0 , \qquad \frac{\partial x}{\partial t}(\eta,0) = x_{1}(\eta)\geq0 \quad \mbox{for } \eta\in \mathbb{H}^{N}, \end{array}\displaystyle \right . $$

(40)

where $\lambda=\lambda(\eta,t)$ is a function defined and measurable in $\mathbb{R}^{2N+1}\times\mathbb{R}^{+}$ and α, $\beta>1$, $q\in(1,2)$, are real parameters.

Definition 5.1

A local weak solution x of the differential inequality (40) in $Q_{T}=\mathbb{R}^{2N+1}\times(0,T)$, with positive initial data $x_{0},x_{1} \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{2N+1})$, is a locally integrable function such that $x \in L^{\beta}(Q_{T},|\eta|_{\mathbb {H}}^{\alpha}\,d\eta \,dt)$ satisfying

$$\begin{aligned}& \int_{Q_{T}} \bigl(-xD^{q}_{t/T}\varphi+ \lambda x\Delta_{\mathbb {H}}\varphi+ |\eta|_{\mathbb{H}}^{\alpha}|x|^{\beta} \varphi+x_{1}(\eta )D^{q-1}_{t/T}\varphi \bigr)\,d \eta \,dt \\& \quad {}+ \int_{\mathbb{R}^{2N+1}}x_{0}(\eta )D^{q-1}_{t/T} \varphi(0)\,d\eta\leq0 \end{aligned}$$

(41)

for any nonnegative test function $\varphi\in C^{2}_{c}(Q_{T})$ such that $\varphi(\cdot,T)=D^{q -1}_{t/T}\varphi(\cdot,T)=0$.

Remark 5.2

As in Definition 3.1, it is assumed that the integrals in (41) are convergent. In Definition 5.1, if $T=+\infty$, the solution is called global.

Theorem 5.3

Let $N\geq1$ and $\beta>1$. Assume that

$$ \alpha>-2 \quad \textit{and}\quad 1< \beta< \frac{q(Q+\alpha)+2}{q(Q-2)+2}, $$

(42)

then there is no weak nontrivial solution x of the system ($\mathrm{FI}_{q}$).

Proof

The proof is based on an appropriate choice of the test function. Suppose the problem (40) has a nontrivial global weak solution x, let T, R, and $\theta>1$ (which will be given later) be three positive reals, let φ be a smooth nonnegative test function, since the initial data $x_{0}$, $x_{1}$ are nonnegative and $D^{q -1}_{t/T}\varphi\geq0$ (from (8)), then the variational formulation (41) implies

$$ \int_{Q_{{TR^{4/\theta}}}} |\eta|_{\mathbb{H}}^{\alpha}|x|^{\beta } \varphi \,d\eta \,dt \leq \int_{Q_{{TR^{4/\theta}}}} xD^{q}_{t/TR^{4/\theta}}\varphi \,d\eta \,dt - \int_{Q_{{TR^{4/\theta}}}} \lambda x\Delta_{\mathbb{H}}\varphi \,d\eta \,dt. $$

(43)

The test function φ should be given to ensure that

$$\int_{Q_{{TR^{4/\theta}}}} \bigl(\bigl\vert D^{q}_{t/T} \varphi\bigr\vert ^{\beta '}+\vert \Delta_{\mathbb{H}}\varphi \vert ^{\beta'} \bigr) \bigl( |\eta |_{\mathbb{H}}^{\alpha}\varphi \bigr)^{-\beta'/\beta}\,d\eta \,dt < \infty. $$

To estimate the right side of (43), we apply Young’s inequality for an arbitrary $\varepsilon>0$, we have

$$\begin{aligned} \int_{Q_{{TR^{4/\theta}}}} xD^{q}_{t/TR^{4/\theta}}\varphi \,d\eta \,dt =& \int_{Q_{{TR^{4/\theta}}}} x \bigl( |\eta|_{\mathbb{H}}^{\alpha } \varphi \bigr)^{\frac{1}{\beta}} \bigl( |\eta|_{\mathbb{H}}^{\alpha } \varphi \bigr)^{-\frac{1}{\beta}} D^{q}_{t/TR^{4/\theta}}\varphi \,d\eta \,dt \\ \leq& \varepsilon \int_{Q_{{TR^{4/\theta}}}} |\eta|_{\mathbb {H}}^{\alpha}|x|^{\beta} \varphi \,d\eta \,dt \\ &{}+ C_{\varepsilon} \int _{Q_{{TR^{4/\theta}}}}\bigl\vert D^{q}_{t/TR^{4/\theta}}\varphi \bigr\vert ^{\beta'} \bigl( |\eta|_{\mathbb{H}}^{\alpha}\varphi \bigr)^{-\frac {\beta'}{\beta}}\,d\eta \,dt \end{aligned}$$

and

$$\begin{aligned} \int_{Q_{{TR^{4/\theta}}}} \lambda x \Delta_{\mathbb{H}}\varphi \,d\eta \,dt =& \int_{Q_{{TR^{4/\theta}}}} \lambda x \bigl( |\eta|_{\mathbb {H}}^{\alpha} \varphi \bigr)^{\frac{1}{\beta}} \bigl( |\eta|_{\mathbb {H}}^{\alpha}\varphi \bigr)^{-\frac{1}{\beta}} \Delta_{\mathbb {H}}\varphi \,d\eta \,dt \\ \leq& \varepsilon \int_{Q_{{TR^{4/\theta}}}} |\eta|_{\mathbb {H}}^{\alpha}|x|^{\beta} \varphi \,d\eta \,dt \\ &{}+ C_{\varepsilon} \Vert \lambda \Vert _{\infty}^{\beta'} \int_{Q_{{TR^{4/\theta}}}} \vert \Delta_{\mathbb{H}}\varphi \vert ^{\beta'} \bigl( |\eta|_{\mathbb {H}}^{\alpha}\varphi \bigr)^{-\frac{\beta'}{\beta}}\,d\eta \,dt. \end{aligned}$$

By considering ε small enough, we have

$$ \int_{Q_{{TR^{4/\theta}}}} |\eta|_{\mathbb{H}}^{\alpha}|x|^{\beta } \varphi \,d\eta \,dt \leq C_{\varepsilon} \int_{Q_{{TR^{4/\theta}}}} \bigl(\bigl\vert D^{q}_{t/TR^{4/\theta}} \varphi\bigr\vert ^{\beta'}+\vert \Delta _{\mathbb{H}}\varphi \vert ^{\beta'} \bigr) \bigl( |\eta|_{\mathbb {H}}^{\alpha}\varphi \bigr)^{-\frac{\beta'}{\beta}} \,d\eta \,dt. $$

(44)

Take

$$\varphi(\eta,t)= \varphi(x,y,\tau,t)=\Phi \biggl(\frac{\tau +|x|^{2}+|y|^{2}+t^{\theta}}{R^{4}} \biggr), $$

where $\Phi\in\mathcal{D}(\mathbb{R}^{+})$, which satisfies $0\leq \Phi\leq1$ and (19), therefore

$$ \Delta_{\mathbb{H}}\varphi(\eta,t)=\frac{4N\Phi'(\rho)}{R^{4}}+\frac{8 \Phi''(\rho)}{R^{8}} \bigl[|x|^{2}+|y|^{2} \bigr], $$

(45)

where

$$\rho=\frac{\tau+|x|^{2}+|y|^{2}+|t|^{\theta}}{R^{4}}. $$

To estimate the right-hand side in (44), we again change the variables,

$$\tilde{t}=R^{-4/\theta}t,\qquad \tilde{\tau}=R^{-4}\tau, \qquad \tilde {x}=R^{-2}x,\qquad \tilde{y}=R^{-2}y, $$

we put

$$\tilde{\rho}=\tilde{\tau}+|\tilde{x}|^{2}+|\tilde {y}|^{2}+\tilde{t}^{\theta}. $$

To guarantee that $supp\Phi\subseteq\Omega$, we assume that

$$\Omega= \bigl\{ (\tilde{\eta},\tilde{t})=(\tilde {x},\tilde{y},\tilde{\tau}, \tilde{t}) \in\mathbb {R}^{2N+1}\times\mathbb{R}, \tilde{\rho} \leq2 \bigr\} . $$

Therefore,

$$ \bigl\vert \Delta_{\mathbb{H}}\varphi(\tilde{\eta},\tilde{t}) \bigr\vert \leq\frac{C}{R^{4}} \quad \forall(\tilde{\eta},\tilde{t}) \in \Omega, $$

(46)

from $d\eta \,dt= R^{4N+4+4/\theta}\,d\tilde{\eta}\,d\tilde{t}$, $|\eta|_{\mathbb{H}}=R^{2} |\tilde{\eta}|_{\mathbb{H}}$, and $\vert D^{q}_{t/TR^{4/\theta}} \varphi \vert =R^{\frac{-4q}{\theta}} \vert D^{q}_{t/T} \varphi \vert$, we have (44) so that

$$\begin{aligned}& \int_{Q_{{TR^{4/\theta}}}} \vert \Delta_{\mathbb{H}}\varphi \vert ^{\beta'} \bigl( |\eta|_{\mathbb{H}}^{\alpha}|x|^{\beta} \bigr)^{-\frac {\beta'}{\beta}} \,d\eta \,dt \\& \quad \leq R^{-4\beta'+4N+4+\frac{4}{\theta }-2\alpha\frac{\beta'}{\beta}} \int_{\Omega} \vert \Delta_{\mathbb {H}}\Phi\circ\tilde{\rho} \vert ^{\beta'} \bigl( |\tilde{\eta }|_{\mathbb{H}}^{\alpha}\Phi \circ\tilde{\rho} \bigr)^{-\frac {\beta'}{\beta}}\,d\tilde{\eta} \,d\tilde{t} \end{aligned}$$

(47)

and

$$\begin{aligned}& \int_{Q_{{TR^{4/\theta}}}}\bigl\vert D^{q}_{t/TR^{4/\theta}} \varphi \bigr\vert ^{\beta'} \bigl( |\eta|_{\mathbb{H}}^{\alpha}|x|^{\beta} \bigr)^{-\frac {\beta'}{\beta}} \,d\eta \,dt \\& \quad \leq R^{-\frac{4q}{\theta}\beta'+4N+4+\frac {4}{\theta}-2\alpha\frac{\beta'}{\beta}} \int_{\Omega}\bigl\vert D^{q}_{t/T} \Phi \circ\tilde{\rho} \bigr\vert ^{\beta'} \bigl( |\tilde{\eta }|_{\mathbb{H}}^{\alpha}\Phi\circ\tilde{\rho} \bigr)^{-\frac {\beta'}{\beta}}\,d \tilde{\eta} \,d\tilde{t}. \end{aligned}$$

(48)

For the same exponent of R in (47) and (48), it is convenient to write $\theta=q$, then

$$ \int_{Q_{{TR^{4/q}}}} |\eta|_{\mathbb{H}}^{\alpha}|x|^{\beta} \varphi \,d\eta \,dt \leq C R^{-4\beta'+4N+4+\frac{4}{q}-2\alpha\frac{\beta'}{\beta}}, $$

(49)

where

$$C=C_{\varepsilon} \int_{\Omega} \bigl(\bigl\vert D^{q}_{t/T} \Phi\circ \tilde{\rho} \bigr\vert ^{\beta'} +\vert \Delta_{\mathbb{H}}\Phi\circ \tilde{\rho} \vert ^{\beta'} \bigr) \bigl(|\tilde{\eta }|_{\mathbb{H}}^{\alpha}\Phi\circ\tilde{\rho} \bigr)^{-\frac {\beta'}{\beta}}\,d\tilde{\eta} \,d\tilde{t}. $$

In the case that

$$1< \beta< \frac{q(Q+\alpha)+2}{q(Q-2)+2}, $$

the exponent of R in (49) is negative, it means that $R\longrightarrow+\infty$ is qualified to apply Fatou’s lemma to get

$$ \int_{0}^{\infty} \int_{\mathbb{R}^{2N+1}}|\eta|_{\mathbb{H}}^{\alpha }|x|^{\beta} \,d\eta \,dt=0. $$

(50)

Thus, $x\equiv0$, and this contradicts the fact that x is a nontrivial solution of (40). □

Remark 5.4

The positivity condition on the initial data can be weakened and replaced by

$$ \int_{Q_{T}}x_{1}(\eta)D^{q-1}_{t/T} \varphi \,d\eta \,dt+ \int_{\mathbb{R}^{2N+1}}x_{0}(\eta)D^{q-1}_{t/T} \varphi(0)\,d\eta\geq0. $$

Remark 5.5

The assertion $\alpha>-2$ and $1<\beta< \frac{q(Q+\alpha )+2}{q(Q-2)+2}$ is equivalent to $Q < 2 (1-\frac{1}{q} )+\frac{\alpha+2}{\beta-1}$, which motivates that Theorem 5.3 is a special case of Theorem 4.2 (in other words $(\mathrm{FI}_{q})\equiv(\mathrm{FS}^{1}_{q})$).

Remark 5.6

$q =2$ covers the case of a hyperbolic inequality of the type

$$ \frac{\partial^{2}x}{\partial t^{2}}-\Delta_{\mathbb{H}}(\lambda x)\geq|\eta|_{\mathbb{H}}^{\alpha}|x|^{\beta} $$

studied by Pohozaev and Véron [3].

Remark 5.7

By assuming $q\rightarrow\infty$, then it is easy to find the well-known critical exponent $\beta_{\infty}=\frac{Q+\alpha}{Q-2}$ for the elliptic inequalities [3, 23].