\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 167, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/167\hfil Blow-up of solutions] {Blow-up of solutions to parabolic inequalities in the Heisenberg group} \author[I. Azman, M. Jleli, B. Samet \hfil EJDE-2015/167\hfilneg] {Ibtehal Azman, Mohamed Jleli, Bessem Samet} \address{Ibtehal Azman \newline Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia} \email{ibtehalazman@yahoo.com} \address{Mohamed Jleli \newline Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia} \email{jleli@ksu.edu.sa} \address{Bessem Samet \newline Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia} \email{bsamet@ksu.edu.sa} \thanks{Submitted May 15, 2015. Published June 17, 2015.} \subjclass[2010]{47J35, 35R03} \keywords{Nonexistence; global solution; differential inequality; \hfil\break\indent Heisenberg group} \begin{abstract} We establish a Fujita-type theorem for the blow-up of nonnegative solutions to a certain class of parabolic inequalities in the Heisenberg group. Our proof is based on a duality argument. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In this article, we establish a Fujita-type theorem for parabolic inequality \begin{equation}\label{p} \begin{gathered} u_t-\operatorname{div}_{\mathbb{H}} A(\vartheta,u,\nabla_{\mathbb{H}} u)+f(\vartheta,u,\nabla_{\mathbb{H}} u) \geq u^q,\quad \text{in } \mathcal{H},\\ u\geq 0, \quad \text{a.e. in } \mathcal{H},\\ u(\vartheta,0)=u_0(\vartheta),\quad \text{in }\mathbb{H}. \end{gathered} \end{equation} Here, $\mathbb{H}$ is the $(2N+1)$-dimensional Heisenberg group, $\mathcal{H}=\mathbb{H}\times (0,\infty)$ and $u_0\in L^1_{\rm loc}(\mathbb{H})$. The operator $A: \mathbb{H}\times \mathbb{R}\times \mathbb{R}^{2N+1}\to \mathbb{R}^{2N+1}$ is a Carath\'eodory function satisfying \begin{equation}\label{AAS} (A(\vartheta,\xi,v),v)\geq c_A|A(\vartheta,\xi,v)|^{m'}, \end{equation} where $c_A>0$, $(\cdot,\cdot)$ is the standard scalar product in $\mathbb{R}^{2N+1}$, $|\cdot|=\sqrt{(\cdot,\cdot)}$, and $m'>1$. The function $f: \mathbb{H}\times \mathbb{R}\times \mathbb{R}^{2N+1}\to \mathbb{R}$ is continuous and satisfies \begin{equation}\label{FAS} f(\vartheta,\xi,v)\leq \lambda |A(\vartheta,\xi,v)|^\sigma, \end{equation} where $\lambda\geq 0$, $\sigma=\frac{m'q}{q+1}$, and $q>\max\{1,m-1\}$, with $m=\frac{m'}{m'-1}$. The proof of our main result is based on a duality argument \cite{MP,MP2,MP3}. First, let us recall some background facts that will be used in this article. The $(2N+1)$-dimensional Heisenberg group $\mathbb{H}$ is the space $\mathbb{R}^{2N+1}$ endowed with the group operation $$ \vartheta\diamond \vartheta'=(x+x',y+y',\tau+\tau'+2(x\cdot y'-x'\cdot y)), $$ for all $\vartheta=(x,y,\tau),\vartheta'=(x',y',\tau')\in \mathbb{R}^N \times \mathbb{R}^N\times \mathbb{R}$, where $\cdot$ denotes the standard scalar product in $\mathbb{R}^N$. This group operation endows $\mathbb{H}$ with the structure of a Lie group. The distance from an element $\vartheta=(x,y,\tau)\in \mathbb{H}$ to the origin is given by $$ |\vartheta|_{\mathbb{H}}=\Big(\tau^2+\Big(\sum_{i=1}^N x_i^2+y_i^2\Big)^2\Big)^{1/4}, $$ where $x=(x_1,\dots,x_N)$ and $y=(y_1,\dots,y_N)$. The Gradient $\nabla_{\mathbb{H}}$ over $\mathbb{H}$ is defined by $$ \nabla_{\mathbb{H}}=(X_1, .., X_N,Y_1,..,Y_N), $$ where for $i=1,\dots,N$, $$ X_i=\partial_{x_i}+2y_i\partial_{\tau}\quad\text{and}\quad Y_i=\partial_{y_i}-2x_i\partial_{\tau}. $$ Let $$ M=\begin{pmatrix} I_N&&0&&2y\\ 0 &&I_N&&-2x\\ \end{pmatrix}, $$ where $I_N$ is the identity matrix of size $N$. Then $$ \nabla_{\mathbb{H}}=M\nabla_{\mathbb{R}^{2N+1}}. $$ A simple computation gives the expression $$ |\nabla_{\mathbb{H}}u|^2=4(|x|^2+|y|^2) (\partial_{\tau}u)^2+\sum_{i=1}^N \Bigr((\partial_{x_i}u)^2+(\partial_{y_i}u)^2+ 4\partial_\tau u(y_i\partial_{x_i}u-x_i\partial_{y_i}u)\Bigr). $$ The divergence operator in $\mathbb{H}$ is defined by $$ \operatorname{div}_{\mathbb{H}}(u)=\operatorname{div}_{\mathbb{R}^{2N+1}}(Mu). $$ For more details on Heisenberg groups and partial differential equations in Heisenberg groups, we refer to \cite{A,B,L,R,T} and references therein. For the proof of our main result, the following inequality will be used several times. \begin{lemma}\label{lem1} Let $a,b,\varepsilon >0$. Then $$ ab\leq \varepsilon a^p+c_\varepsilon b^{p'}, $$ where $p>1$, $p'$ is its corresponding conjugate exponent, i.e., $ \frac{1}{p}+\frac{1}{p'}=1$; and $c_\varepsilon= \big(\frac{1}{\varepsilon p}\big)^{p'/p}\frac{1}{p'}$. \end{lemma} \section{Main result} \begin{definition}\label{DF} \rm Let $u\in W^{1,m}_{\rm loc}(\mathcal{H};\mathbb{R}_+)\cap L^q_{\rm loc}(\mathcal{H};\mathbb{R}_+)$ and $u_0\in L^1_{\rm loc}(\mathbb{H};\mathbb{R}_+)$. We say that $u$ is a global weak solution of \eqref{p} if the following conditions are satisfied: \begin{itemize} \item[(i)] $A(\vartheta,u,\nabla_{\mathbb{H}} u)\in L^{m'}_{\rm loc} (\mathcal{H};\mathbb{R}^{2N+1})$; \item[(ii)] For any $\varphi\in W^{1,m}_{\rm loc}(\mathcal{H};\mathbb{R}_+)$ with a compact support, \begin{equation}\label{WS} \begin{aligned} \int_{\mathcal{H}} u^q\varphi\,d\mathcal{H} &\leq \int_{\mathcal{H}} (A(\vartheta,u,\nabla_{\mathbb{H}} u), \nabla_{\mathbb{H}}\varphi)\,d\mathcal{H} +\int_{\mathcal{H}} f(\vartheta,u,\nabla_{\mathbb{H}} u)\varphi\,d\mathcal{H}\\ &\quad - \int_{\mathcal{H}} u\varphi_{t}\,d\mathcal{H} - \int_{\mathbb{H}}u_0(\vartheta) \varphi(\vartheta,0)\,d\vartheta. \end{aligned} \end{equation} \end{itemize} \end{definition} Observe that all the integrals in \eqref{WS} are well defined. Our main result is given in the following theorem. \begin{theorem}\label{thm1} Assume that conditions \eqref{AAS} and \eqref{FAS} are satisfied. Let us consider $\alpha\in (\alpha_0,0)$, where $\alpha_0=\max\{-1,1-m\}<0$. If \begin{equation}\label{lam} 0\leq \lambda<\lambda^*=(q+1)\Big(\frac{|\alpha|c_A}{q}\Big)^{\frac{q}{q+1}} \end{equation} and \begin{equation}\label{cr} \max\{1,m-1\}0$, where $r=\frac{q+\alpha}{1+\alpha}$, and $s=\frac{q+\alpha}{q-m+1}$. \end{lemma} \begin{proof} Let $\varepsilon>0$ be fixed and $\alpha \in (\alpha_0,0)$. Suppose that $u$ is a global weak solution to \eqref{p}. Let $$ u_\varepsilon(\vartheta,t)= u(\vartheta,t)+\varepsilon,\quad (\vartheta,t)\in \mathcal{H}. $$ Define $\varphi_\varepsilon$ as $$ \varphi_\varepsilon(\vartheta,t) =u_\varepsilon^\alpha(\vartheta,t)\varphi(\vartheta,t), $$ where $\varphi\in W^{1,\infty}(\mathcal{H};\mathbb{R}_+)$ has a compact support. Observe that $\varphi_\varepsilon$ belongs to the set of admissible test functions in the sense of Definition \ref{DF}. By \eqref{WS}, we have \begin{equation}\label{ta} \begin{aligned} &\int_{\mathcal{H}} u^q u_\varepsilon^\alpha\varphi\,d\mathcal{H} \\ &\leq \alpha \int_{\mathcal{H}} (A(\vartheta,u,\nabla_{\mathbb{H}} u),\nabla_{\mathbb{H}} u)u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H} + \int_{\mathcal{H}} (A(\vartheta,u,\nabla_{\mathbb{H}} u),\nabla_{\mathbb{H}} \varphi)u_\varepsilon^{\alpha}\,d\mathcal{H} \\ &\quad +\int_{\mathcal{H}} f(\vartheta,u,\nabla_{\mathbb{H}} u)u_\varepsilon^\alpha \varphi\,d\mathcal{H} -\frac{1}{\alpha+1}\int_{\mathcal{H}} u_\varepsilon^{\alpha+1}\varphi_{t}\,d\mathcal{H}\\ &\quad -\frac{1}{\alpha+1}\int_{\mathbb{H}}(u_0(\vartheta) +\varepsilon)^{\alpha+1} \varphi(\vartheta,0)\,d\vartheta. \end{aligned} \end{equation} Using the condition \eqref{AAS}, we obtain $$ \int_{\mathcal{H}} (A(\vartheta,u,\nabla_{\mathbb{H}} u),\nabla_{\mathbb{H}} u) u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H} \geq c_A \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^{m'} u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H}. $$ Since $\alpha<0$, we have \begin{equation}\label{esA} \alpha \int_{\mathcal{H}} (A(\vartheta,u,\nabla_{\mathbb{H}} u), \nabla_{\mathbb{H}} u)u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H} \leq c_A \alpha \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^{m'} u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H}. \end{equation} The Cauchy-Schwarz inequality yields \begin{equation}\label{esA2} \int_{\mathcal{H}} (A(\vartheta,u,\nabla_{\mathbb{H}} u), \nabla_{\mathbb{H}}\varphi)u_\varepsilon^{\alpha}\,d\mathcal{H} \leq \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)| |\nabla_{\mathbb{H}}\varphi| u_\varepsilon^{\alpha}\,d\mathcal{H}. \end{equation} Using the condition \eqref{FAS}, we obtain \begin{equation}\label{esF} \int_{\mathcal{H}} f(\vartheta,u,\nabla_{\mathbb{H}} u)u_\varepsilon^\alpha \varphi\,d\mathcal{H} \leq \lambda \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^\sigma u_\varepsilon^\alpha \varphi\,d\mathcal{H}. \end{equation} Now, \eqref{ta}, \eqref{esA}, \eqref{esA2} and \eqref{esF} yield \begin{equation}\label{OK} \begin{aligned} &\int_{\mathcal{H}} u^q u_\varepsilon^\alpha\varphi\,d\mathcal{H} + c_A|\alpha| \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u) |^{m'}u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H} \\ &+ \frac{1}{\alpha+1}\int_{\mathbb{H}}(u_0(\vartheta) +\varepsilon)^{\alpha+1} \varphi(\vartheta,0)\,d\vartheta\\ &\leq \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)| |\nabla_{\mathbb{H}}\varphi| u_\varepsilon^{\alpha}\,d\mathcal{H} +\lambda \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u) |^\sigma u_\varepsilon^\alpha \varphi\,d\mathcal{H} \\ &\quad + \frac{1}{\alpha+1}\int_{\mathcal{H}} u_\varepsilon^{\alpha+1}|\varphi_{t}|\,d\mathcal{H}. \end{aligned} \end{equation} Now, using Lemma \ref{lem1}, we estimate the individual terms on the right hand side of \eqref{OK}. \smallskip \noindent $\bullet$ Estimation of $\int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)| |\nabla_{\mathbb{H}}\varphi| u_\varepsilon^{\alpha}\,d\mathcal{H}$. We have $$ |A(\vartheta,u,\nabla_{\mathbb{H}} u)||\nabla_{\mathbb{H}}\varphi| u_\varepsilon^{\alpha} =\Big(|A(\vartheta,u,\nabla_{\mathbb{H}} u)| u_\varepsilon^{\frac{\alpha-1}{m'}}\varphi^{\frac{1}{m'}}\Big) \Big(u_\varepsilon^{\frac{\alpha+m-1}{m}}\varphi^{\frac{-1}{m'}} |\nabla_{\mathbb{H}}\varphi|\Big). $$ Applying Lemma \ref{lem1} with parameters $m'$ and $m$, we obtain \begin{align*} &\int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)| |\nabla_{\mathbb{H}}\varphi| u_\varepsilon^{\alpha}\,d\mathcal{H} \\ &\leq \varepsilon_1 \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u) |^{m'} u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H}+c_{\varepsilon_1} \int_{\mathcal{H}} u_\varepsilon^{\alpha+m-1}\varphi^{1-m}|\nabla_{\mathbb{H}}\varphi|^m\,d\mathcal{H}, \end{align*} for some $\varepsilon_1>0$. Again, writing $$ u_\varepsilon^{\alpha+m-1}\varphi^{1-m}|\nabla_{\mathbb{H}}\varphi|^m =\left(\varphi^{\frac{1-ms}{s}}|\nabla_{\mathbb{H}}\varphi|^m\right) \left(\varphi^{\frac{s-1}{s}}u_\varepsilon^{\alpha+m-1}\right) $$ and using Lemma \ref{lem1} with parameters $s$ and $s'$, we obtain $$ \int_{\mathcal{H}} u_\varepsilon^{\alpha+m-1}\varphi^{1-m}|\nabla_{\mathbb{H}}\varphi|^m\,d\mathcal{H}\leq \varepsilon_2 \int_{\mathcal{H}} \varphi^{1-ms} |\nabla_{\mathbb{H}}\varphi|^{ms}\,d\mathcal{H}+c_{\varepsilon_2} \int_{\mathcal{H}} \varphi u_\varepsilon^{(\alpha+m-1)s'}\,d\mathcal{H}, $$ for some $\varepsilon_2>0$. As consequence, we have \begin{equation}\label{es1} \begin{aligned} &\int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)| |\nabla_{\mathbb{H}}\varphi| u_\varepsilon^{\alpha}\,d\mathcal{H} \\ &\leq \varepsilon_1 \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^{m'} u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H}\\ &\quad +c_{\varepsilon_1}\varepsilon_2 \int_{\mathcal{H}} \varphi^{1-ms} |\nabla_{\mathbb{H}}\varphi|^{ms}\,d\mathcal{H} +c_{\varepsilon_1}c_{\varepsilon_2} \int_{\mathcal{H}} \varphi u_\varepsilon^{(\alpha+m-1)s'}\,d\mathcal{H}. \end{aligned} \end{equation} \noindent $\bullet$ Estimation of $\int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^\sigma u_\varepsilon^\alpha \varphi\,d\mathcal{H}$. We write $$ |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^\sigma u_\varepsilon^\alpha \varphi =\Big( |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^\sigma u_\varepsilon^{\frac{(\alpha-1)\sigma}{m'}}\varphi^{\frac{\sigma}{m'}}\Big) \Big(\varphi^{\frac{m'-\sigma}{m'}} u_\varepsilon^{\frac{\alpha m' -(\alpha-1)\sigma}{m'}}\Big). $$ We apply Lemma \ref{lem1} with parameters $\frac{m'}{\sigma}$ and $\frac{m'}{m'-\sigma}$ to obtain \begin{align*} &\int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^\sigma u_\varepsilon^\alpha \varphi\,d\mathcal{H}\\ &\leq \varepsilon_3 \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^{m'} u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H}+c_{\varepsilon_3} \int_{\mathcal{H}} \varphi u_\varepsilon^{\alpha-1+\frac{m'}{m'-\sigma}}\,d\mathcal{H}, \end{align*} for some $\varepsilon_3>0$. Since $\sigma=m'q/(q+1)$, we obtain \begin{equation}\label{es2} \begin{aligned} &\int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^\sigma u_\varepsilon^\alpha \varphi\,d\mathcal{H}\\ &\leq \varepsilon_3 \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u) |^{m'} u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H} +c_{\varepsilon_3} \int_{\mathcal{H}} \varphi u_\varepsilon^{\alpha+q}\,d\mathcal{H}. \end{aligned} \end{equation} \noindent $\bullet$ Estimation of $\int_{\mathcal{H}} u_\varepsilon^{\alpha+1}|\varphi_{t}|\,d\mathcal{H}$. Similarly, we write $$ u_\varepsilon^{\alpha+1}|\varphi_{t}| =\left(u_\varepsilon^{\alpha+1}\varphi^{\frac{1}{r}}\right) \left(\varphi^{-\frac{1}{r}}|\varphi_{t}|\right). $$ Lemma \ref{lem1} with parameters $r$ and $r'$ yields \begin{equation}\label{es3} \int_{\mathcal{H}} u_\varepsilon^{\alpha+1}|\varphi_{t}|\,d\mathcal{H} \leq \varepsilon_4 \int_{\mathcal{H}}u_\varepsilon^{(\alpha+1)r}\varphi \,d\mathcal{H} +c_{\varepsilon_4}\int_{\mathcal{H}}\Big(\frac{ |\varphi_t|^r}{\varphi}\Big)^{\frac{1}{r-1}}\,d\mathcal{H}, \end{equation} for some $\varepsilon_4>0$. Now, substituting \eqref{es1}, \eqref{es2} and \eqref{es3} in \eqref{OK}, we obtain \begin{align*} &\int_{\mathcal{H}} u^q u_\varepsilon^\alpha\varphi\,d\mathcal{H} + c_A|\alpha| \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^{m'} u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H} \\ &+ \frac{1}{\alpha+1}\int_{\mathbb{H}}(u_0(\vartheta)+\varepsilon)^{\alpha+1} \varphi(\vartheta,0)\,d\vartheta\\ & \leq \varepsilon_1 \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^{m'} u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H} +c_{\varepsilon_1}\varepsilon_2 \int_{\mathcal{H}} \varphi^{1-ms} |\nabla_{\mathbb{H}}\varphi|^{ms}\,d\mathcal{H}\\ &\quad +c_{\varepsilon_1}c_{\varepsilon_2} \int_{\mathcal{H}} \varphi u_\varepsilon^{(\alpha+m-1)s'}\,d\mathcal{H} +\lambda \varepsilon_3 \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^{m'} u_\varepsilon^{\alpha-1}\varphi\,d\mathcal{H}\\ &\quad +\lambda c_{\varepsilon_3} \int_{\mathcal{H}} \varphi u_\varepsilon^{\alpha+q}\,d\mathcal{H} +\frac{\varepsilon_4}{\alpha+1} \int_{\mathcal{H}}u_\varepsilon^{(\alpha+1)r}\varphi \,d\mathcal{H} +\frac{c_{\varepsilon_4}}{\alpha+1}\int_{\mathcal{H}}\Big(\frac{ |\varphi_t|^r}{\varphi}\Big)^{\frac{1}{r-1}}\,d\mathcal{H}. \end{align*} Now, we let $\varepsilon\to 0$ in the obtained inequality, we use Fatou's lemma and the dominated convergence theorem to obtain \begin{align*} &\int_{\mathcal{H}} u^{q+\alpha}\varphi\,d\mathcal{H} + c_A|\alpha| \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^{m'} u^{\alpha-1}\varphi\,d\mathcal{H} + \frac{1}{\alpha+1}\int_{\mathbb{H}}u_0^{\alpha+1}(\vartheta) \varphi(\vartheta,0)\,d\vartheta\\ & \leq \varepsilon_1 \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^{m'} u^{\alpha-1}\varphi\,d\mathcal{H} +c_{\varepsilon_1}\varepsilon_2 \int_{\mathcal{H}} \varphi^{1-ms} |\nabla_{\mathbb{H}}\varphi|^{ms}\,d\mathcal{H}\\ &\quad +c_{\varepsilon_1}c_{\varepsilon_2} \int_{\mathcal{H}} \varphi u^{q+\alpha}\,d\mathcal{H} +\lambda \varepsilon_3 \int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u) |^{m'} u^{\alpha-1}\varphi\,d\mathcal{H} \\ &\quad +\lambda c_{\varepsilon_3} \int_{\mathcal{H}} \varphi u^{\alpha+q}\,d\mathcal{H} +\frac{\varepsilon_4}{\alpha+1} \int_{\mathcal{H}}u^{q+\alpha}\varphi \,d\mathcal{H} +\frac{c_{\varepsilon_4}}{\alpha+1}\int_{\mathcal{H}}\Big(\frac{ |\varphi_t|^r}{\varphi}\Big)^{\frac{1}{r-1}}\,d\mathcal{H}, \end{align*} i.e., \begin{equation} \label{TB} \begin{aligned} &\mathcal{A}\int_{\mathcal{H}} u^{q+\alpha}\varphi\,d\mathcal{H} +\mathcal{B}\int_{\mathcal{H}} |A(\vartheta,u,\nabla_{\mathbb{H}} u)|^{m'}u^{\alpha-1} \varphi\,d\mathcal{H}\\ &+\frac{1}{\alpha+1}\int_{\mathbb{H}}u_0^{\alpha+1}(\vartheta) \varphi(\vartheta,0)\,d\vartheta\\ &\leq \max\big\{c_{\varepsilon_1}\varepsilon_2, \frac{c_{\varepsilon_4}}{\alpha+1}\big\} \Big(\int_{\mathcal{H}} \varphi^{1-ms} |\nabla_{\mathbb{H}}\varphi|^{ms}\,d\mathcal{H} +\int_{\mathcal{H}}\Big(\frac{|\varphi_t|^r}{\varphi}\Big)^{\frac{1}{r-1}}\,d\mathcal{H}\Big), \end{aligned} \end{equation} where $$ \mathcal{A}=1-\lambda c_{\varepsilon_3}-\frac{\varepsilon_4}{\alpha+1} -c_{\varepsilon_1}c_{\varepsilon_2}\quad\text{and}\quad \mathcal{B}=c_A|\alpha|-\varepsilon_1-\lambda \varepsilon_3. $$ From Lemma \ref{lem1}, we have \begin{gather*} c_{\varepsilon_1}=\frac{1}{m}\Big(\frac{1}{\varepsilon_1m'}\Big)^{m'/m},\quad c_{\varepsilon_2}=\frac{1}{s'}\Big(\frac{1}{\varepsilon_2s}\Big)^{s'/s},\\\ c_{\varepsilon_3}=\frac{1}{q+1}\Big(\frac{q}{\varepsilon_3(q+1)}\Big)^{q},\quad c_{\varepsilon_4}=\frac{1}{r'}\Big(\frac{1}{\varepsilon_4r}\Big)^{r'/r}. \end{gather*} For $\varepsilon_1>0$ small enough, taking \begin{equation}\label{lam1} 0\leq \lambda<\frac{c_A|\alpha|}{\varepsilon_3}, \end{equation} we obtain $\mathcal{B}>0$. For $\varepsilon_i>0$ small enough ($i=1,2,4$), taking \begin{equation}\label{lam2} 0\leq \lambda<\frac{1}{c_{\varepsilon_3}}, \end{equation} we obtain $\mathcal{A}>0$. Now, we choose $\varepsilon_3>0$ such that $$ \frac{c_A|\alpha|}{\varepsilon_3}=\frac{1}{c_{\varepsilon_3}}, $$ i.e., $$ \frac{c_A|\alpha|}{\varepsilon_3} =(q+1)\Big(\frac{q}{\varepsilon_3(q+1)}\Big)^{-q}. $$ A simple computation yields $$ \varepsilon_3=(c_A|\alpha|)^{\frac{1}{q+1}} \Big(\frac{q}{q+1}\Big)^{\frac{q}{q+1}}\Big(\frac{1}{q+1}\Big)^{\frac{1}{q+1}}. $$ We substitute $\varepsilon_3$ into \eqref{lam1} (or \eqref{lam2}) to get $$ 0\leq \lambda <\lambda^*=(q+1)\Big(\frac{|\alpha|c_A}{q}\Big)^{\frac{q}{q+1}}. $$ Thus, for $0\leq \lambda<\lambda^*$ and $\varepsilon_i>0$ small enough ($i=1,2,4$), we have \begin{equation}\label{signe} \mathcal{A}>0\quad\text{and}\quad \mathcal{B}>0. \end{equation} Finally, the desired result follows from \eqref{TB} and \eqref{signe} with $$ C=\frac{\max\{c_{\varepsilon_1}\varepsilon_2, \frac{c_{\varepsilon_4}}{\alpha+1}\}} {\min\{\mathcal{A},\mathcal{B},\frac{1}{\alpha+1}\}}. $$ The lemma is proved. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1}] Suppose that $u$ is a nontrivial global weak solution to \eqref{p}. Let us consider the test function \[ \varphi_R(\vartheta,t)=\varphi_R(x,y,\tau,t)= \phi^\omega\Big(\frac{t^{2\theta_1}+|x|^{4\theta_2}+|y|^{4\theta_2} +\tau^{2\theta_2}}{R^{4\theta_2}}\Big),\quad R>0, \; \omega\gg 1, \] where $\phi\in C_0^\infty(\mathbb{R}^+)$ is a decreasing function satisfying $$ \phi(z)=\begin{cases} 1 &\text{if } 0\leq z\leq 1\\ 0 &\text{if } z\geq 2, \end{cases} $$ and $\theta_j$, $j=1,2$ are positive parameters, whose exact values will be specified later. Let $$ \rho=\frac{t^{2\theta_1}+|x|^{4\theta_2}+|y|^{4\theta_2} +\tau^{2\theta_2}}{R^{4\theta_2}}\,. $$ Clearly $\varphi_R$ is supported on $$ \Omega_R=\{(\vartheta,t)\in \mathcal{H}:\,0\leq \rho\leq 2\}, $$ while $(\varphi_R)_{t}$ and $\nabla_{\mathbb{H}}\varphi_R$ are supported on $$ \Theta_R=\{(\vartheta,t)\in \mathcal H:\, 1\leq \rho\leq 2\}. $$ A simple computation yields \[ \partial_{t}\varphi_R(\vartheta,t)=2\theta_1\omega t^{2\theta_1-1} R^{-4\theta_2} \phi^{\omega-1}(\rho) \phi'(\rho), \] while \begin{align*} \nabla_{\mathbb H} \varphi_R(t,\vartheta)|^2 &= 16 \theta_2^2\omega^2 R^{-8\theta_2}(\phi'(\rho))^2\phi^{2\omega-2}(\rho) \Big( (|x|^2+|y|^2)\tau^{4\theta_2-2}\\ &\quad +(|x|^{8\theta_2-2}+|y|^{8\theta_2-2}) +2\tau^{2\theta_2-1}\sum_{i=1}^Nx_iy_i(|x|^{4\theta_2-2} -|y|^{4\theta_2-2})\Big). \end{align*} Then, for all $(\vartheta,t)\in \Omega_R$, we have \begin{equation}\label{V} R|\nabla_{\mathbb H} \varphi_R|+R^{2\theta_2/\theta_1}|\partial_{t}\varphi_R| \leq C |\phi'(\rho)|\phi^{\omega-1}(\rho). \end{equation} For simplicity, in the sequel, we will write $\varphi$ in the place of $\varphi_R$. Let us consider now the change of variables \[ (x,y,\tau,t)=(\vartheta,t)\mapsto (\widetilde{x},\widetilde{y},\widetilde{\tau},\widetilde{t})=(\widetilde{\vartheta},\widetilde{t}), \] where $$ \widetilde{t}=R^{-2\theta_2/\theta_1}t, \quad \widetilde{x}=R^{-1}x,\quad \widetilde{y}=R^{-1}y,\quad \widetilde{\tau}=R^{-2}\tau. $$ In the same way, let \begin{gather*} \widetilde{\rho}=\widetilde{t}^{2\theta_1}+|\widetilde{x}|^{4\theta_2} +|\widetilde{y}|^{4\theta_2}+\widetilde{\tau}^{2\theta_2},\\ \widetilde{\Omega}=\{(\widetilde{x},\widetilde{y},\widetilde{\tau}, \widetilde{t})\in \mathcal{H}: 0\leq \widetilde{\rho}\leq 2\},\\ \widetilde{\Theta}= \{(\widetilde{x},\widetilde{y},\widetilde{\tau}, \widetilde{t})\in \mathcal{H}: 1\leq \widetilde{\rho}\leq 2\}. \end{gather*} Using the above change of variables together with \eqref{V}, we obtained \begin{equation}\label{JJ1} \int_\mathcal{H} \Bigr(\frac{|\varphi_{t}|^r}{\varphi}\Bigr)^\frac{1}{r-1} \,d\,\mathcal{H} \leq C R^{Q+2\frac{\theta_2}{\theta_1}(1-\frac{r}{r-1})} \int_\mathcal{H}\phi^{\omega-\frac{r}{r-1}} |\phi'|^{\frac{r}{r-1}} \,d\widetilde{\mathcal{H}} \end{equation} and \begin{equation}\label{JJ2} \int_\mathcal{H} \varphi^{1-ms}|\nabla_{\mathbb{H}} \varphi|^{ms} \,d\mathcal{H} \leq C R^{Q+2\frac{\theta_2}{\theta_1}-ms} \int_\mathcal{H}\phi^{\omega-ms}|\phi'|^{ms}\,d\widetilde{\mathcal{H}}. \end{equation} Setting $$ \frac{\theta_2}{\theta_1}=\frac{ms(r-1)}{2r}, $$ we have \begin{equation}\label{GG} Q+2\frac{\theta_2}{\theta_1}(1-\frac{r}{r-1})= Q+2\frac{\theta_2}{\theta_1}-ms=Q-\frac{m(q+\alpha)}{q-m+1}+\frac{m(q-1)}{q-m+1}. \end{equation} Using \eqref{SI}, \eqref{JJ1}-\eqref{GG}, we obtain \begin{equation}\label{EEG} \int_{\mathcal{H}}u^{q+\alpha} \varphi \,d\mathcal{H}\leq C R^{Q-\frac{m(q+\alpha)}{q-m+1}+\frac{m(q-1)k}{q-m+1}}. \end{equation} Furthermore, noting that $$ Q-\frac{m(q+\alpha)}{q-m+1}+\frac{m(q-1)}{q-m+1}<0 $$ for $q< m-1+\frac{m}{Q}$ and some $\alpha \in (\alpha_0,0)$ small enough. Under the above condition, letting $R\to \infty$ in \eqref{EEG} and using the monotone convergence theorem, we obtain $$ \int_{\mathcal{H}}u^{q+\alpha} \,d\mathcal{H}\leq 0, $$ which contradicts our assumption about $u$. This completes the proof. \end{proof} Let us consider now some examples where Theorem \ref{thm1} can be applied. \begin{corollary}\label{coro1} If $\max\{1,m-1\}