Nonexistence results for pseudo-parabolic equations in the Heisenberg group

Abstract

We derive sufficient conditions for the nonexistence of global weak solutions to the nonlinear pseudo-parabolic equation

$$\begin{aligned} u_t-\Delta _{\mathbb H}u_t-\Delta _{\mathbb H}u=|u|^p+f(t,\vartheta ), \quad (t,\vartheta )\in (0,\infty )\times \mathbb {H}, \end{aligned}$$

where $\Delta _\mathbb {H}$ is the Kohn–Laplace operator on the $(2N+1)$-dimensional Heisenberg group $\mathbb {H}$, $p>1$ and $f(t,\vartheta )$ is a given function. Next, we extend this result to the case of systems. Our technique of proof is based on the test function method.

1 Introduction

In this paper, we are concerned with nonexistence of global weak solutions to the nonlinear pseudo-parabolic equation

$$\begin{aligned} u_t-\Delta _{\mathbb H}u_t-\Delta _{\mathbb H}u=|u|^p+f(t,\vartheta ), \quad (t,\vartheta )\in (0,\infty )\times \mathbb {H}, \end{aligned}$$

(1.1)

under the initial condition

$$\begin{aligned} u(0,\vartheta )=u_0(\vartheta ),\,\quad \vartheta \in \mathbb {H}, \end{aligned}$$

(1.2)

where $\Delta _\mathbb {H}$ is the Kohn–Laplace operator on the $(2N+1)$-dimensional Heisenberg group $\mathbb {H}$, $p>1$ and $f(t,\vartheta )$ is a given function. In the Euclidean case, this type of equations models a variety of important physical process, for example, the seepage of homogeneous fluids through a fissured rock [2], the unidirectional propagation of nonlinear dispersive long waves [3] and the aggregation of populations [13]. Furthermore, problem (1.1) and (1.2) can be regarded as a Sobolev type equation or Sobolev–Galpern type equation [16].

The critical Fujita exponent to the pseudo-parabolic equation (1.1) and (1.2) (with $f\equiv 0$) in the Euclidean case was determined as $p^*=1+\frac{2}{N}$ in recent years, i.e., by Kaikina et al. [10] for $p>p^*$ and Cao et al. [5] for $p\le p^*$. Yang et al. [17] extended the above results to the case of coupled nonlinear pseudo-parabolic equations.

In this paper, first we provide a sufficient condition for the nonexistence of global weak solutions to the nonlinear problem (1.1) and (1.2). Next, we extend this result to the case of $2\times 2$ systems. More precisely, we consider two kinds of coupled nonlinear pseudo-parabolic equations. First, we consider the system

$$\begin{aligned} \left\{ \begin{array}{lll} u_t-\Delta _{\mathbb H}u_t-\Delta _{\mathbb H}u=|v|^q+f(t,\vartheta ), &{}\quad (t,\vartheta )\in (0,\infty )\times \mathbb {H},\\ v_t-\Delta _{\mathbb H}v_t-\Delta _{\mathbb H}v=|u|^p+g(t,\vartheta ),&{}\quad (t,\vartheta )\in (0,\infty )\times \mathbb {H},\\ u(0,\vartheta )=u_0(\vartheta ),\,\, v(0,\vartheta )=v_0(\vartheta ),&{}\quad \vartheta \in \mathbb {H}, \end{array} \right. \end{aligned}$$

(1.3)

where $p,q>1$ and f, g are given functions, for which we provide a sufficient condition for the nonexistence of global weak solutions. Note that in the Euclidean case, Yang et al. [17] proved that the critical Fujita curve for this system (with $f=g\equiv 0$) is given by $(pq)^*=1+\frac{2}{N}\max \{p+1,q+1\}$. Next, we consider the system

$$\begin{aligned} \left\{ \begin{array}{ll} u_t-\Delta _{\mathbb H}u_t-\Delta _{\mathbb H}v=|v|^q+f(t,\vartheta ),&{}\quad (t,\vartheta )\in (0,\infty )\times \mathbb {H},\\ v_t-\Delta _{\mathbb H}v_t-\Delta _{\mathbb H}u=|u|^p+g(t,\vartheta ),&{}\quad (t,\vartheta )\in (0,\infty )\times \mathbb {H},\\ u(0,\vartheta )=u_0(\vartheta ),\,\, v(0,\vartheta )=v_0(\vartheta ),&{}\quad \vartheta \in \mathbb {H}, \end{array} \right. \end{aligned}$$

(1.4)

where $p,q>1$ and f, g are given functions.

Before stating and proving our main results, let us recall some mathematical preliminaries used here.

The $(2N+1)$-dimensional Heisenberg group $\mathbb {H}$ is the space $\mathbb {R}^{2N+1}$ endowed with the group operation

$$\begin{aligned} \vartheta \diamond \vartheta '=(x+x',y+y',\tau +\tau '+2(x\cdot y'-x'\cdot y)), \end{aligned}$$

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 $\vartheta =(x,y,\tau )\in \mathbb {H}$ to the origin is given by

$$\begin{aligned} |\vartheta |_\mathbb {H}=\left( \tau ^2+\left( \sum _{i=1}^N\left( x_i^2+y_i^2\right) \right) ^2\right) ^{1/4}, \end{aligned}$$

where $x=(x_1,\ldots ,x_N)$ and $y=(y_1,\ldots ,y_N)$.

The Laplacian $\Delta _{\mathbb {H}}$ over $\mathbb {H}$ can be defined from the vectors fields

$$\begin{aligned} X_i=\partial _{x_i}+2y_i\partial _{\tau }\quad \text{ and }\quad Y_i=\partial _{y_i}-2x_i\partial _{\tau }, \end{aligned}$$

for $i=1,\ldots ,N$, as follows

$$\begin{aligned} \Delta _{\mathbb {H}}=\sum _{i=1}^N (X_i^2+Y_i^2), \end{aligned}$$

that is,

$$\begin{aligned} \Delta _{\mathbb {H}}u=\sum _{i=1}^N (\partial ^2_{x_ix_i}u+\partial _{y_iy_i}^2u+4y_i\partial ^2_{x_i\tau }u-4x_i\partial ^2_{y_i\tau }u+4(x_i^2+y_i)^2 \partial ^2_{\tau \tau }u). \end{aligned}$$

For all $\vartheta ,\vartheta '\in \mathbb {H}$, we have

$$\begin{aligned} \Delta _{\mathbb {H}}(u(\vartheta \diamond \vartheta '))=\Delta _{\mathbb {H}}u(\vartheta \diamond \vartheta '). \end{aligned}$$

For $\lambda \in \mathbb {R}$ and $(x,y,\tau )\in \mathbb {H}$, we have

$$\begin{aligned} \Delta _{\mathbb {H}}(u(\lambda x,\lambda y,\lambda ^2\tau ))=\lambda ^2 (\Delta _\mathbb {H}u)(\lambda x,\lambda y,\lambda ^2\tau ). \end{aligned}$$

If $u(\vartheta )=v(|\vartheta |_\mathbb {H})$, then

$$\begin{aligned} \Delta _{\mathbb {H}}v(\rho )=a(\vartheta )\left( \frac{d^2v}{d\rho ^2}+\frac{Q-1}{\rho }\frac{dv}{d\rho }\right) , \end{aligned}$$

where $\rho =|\vartheta |_\mathbb {H}$, $a(\vartheta )=\rho ^{-2}\sum _{i=1}^N (x_i^2+y_i^2)$ and $Q=2N+2$ is the homogeneous dimension of $\mathbb {H}$.

For more details on Heisenberg groups, we refer to [4, 8, 12]. For other nonexistence results in Heisenberg groups, we refer to [1, 4, 6, 7, 9, 14, 15, 18].

2 Results and proofs

Let $\mathcal H=(0,\infty )\times \mathbb {H}$. For $R>0$, let

$$\begin{aligned} \mathcal {U}_R=\{(t,x,y,\tau )\in \mathcal {H}:\,0\le t^2+|x|^4+|y|^4+\tau ^2\le R^4\}. \end{aligned}$$

2.1 The case of a single equation

Let $f\in L^1_{loc}(\mathcal H)$. The definition of solutions we adopt for (1.1) and (1.2) is:

Definition 2.1

We say that u is a global weak solution to (1.1) and (1.2) on $\mathcal {H}$ with initial data $u(0,\cdot )=u_0\in L^1_{loc}(\mathbb {H})$, if $u\in L^{p}_{loc}(\mathcal H)$ and satisfies

$$\begin{aligned}&\int _{\mathcal H} |u|^p\varphi \,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi (0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi \,dtd\vartheta \\&\quad =-\int _{\mathcal H} u\varphi _t\,dt d\vartheta +\int _{\mathcal H} u (\Delta _{\mathbb H}\varphi )_t\,dtd\vartheta -\int _{\mathcal H}u \Delta _{\mathbb {H}}\varphi \,dtd\vartheta \\&\qquad +\,\int _{\mathbb {H}}u_0(\vartheta )\Delta _{\mathbb H}\varphi (0,\vartheta )\,d\vartheta , \end{aligned}$$

for any regular test function $\varphi$, $\varphi (\cdot ,t)=0$, $t\ge T$ (t is large enough).

Our first main result is given by the following theorem.

Theorem 2.2

Let $u_0\in L^1(\mathbb {H})$ and $f^-\in L^1(\mathcal H)$, where $f^-=\max \{-f,0\}$. Suppose that

$$\begin{aligned} \int _{\mathbb {H}}u_0\,d\vartheta +\liminf _{R\rightarrow \infty }\int _{\mathcal {U}_R} f\,dtd\vartheta > 0. \end{aligned}$$

(2.1)

If

$$\begin{aligned} 1<p\le p^*=1+\frac{2}{Q}, \end{aligned}$$

then (1.1) and (1.2) does not admit any global weak solution.

Proof

Suppose that u is a global weak solution to (1.1) and (1.2). Then for any regular test function $\varphi$, we have

$$\begin{aligned}&\int _{\mathcal H} |u|^p\varphi \,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi (0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi \,dtd\vartheta \nonumber \\&\quad \le \int _{\mathcal H} |u| |\varphi _t|\,dt d\vartheta +\int _{\mathcal H} |u| |(\Delta _{\mathbb H}\varphi )_t|\,dtd\vartheta +\int _{\mathcal H}|u| |\Delta _{\mathbb {H}}\varphi |\,dtd\vartheta \nonumber \\&\qquad +\int _{\mathbb {H}}|u_0(\vartheta )| |\Delta _{\mathbb H}\varphi (0,\vartheta )|\,d\vartheta . \end{aligned}$$

(2.2)

Using the $\varepsilon$-Young inequality with parameters p and $p/(p-1)$, we obtain

$$\begin{aligned} \int _{\mathcal H} |u| |\varphi _t|\,dt d\vartheta \le \varepsilon \int _{\mathcal H} |u|^p\varphi \,dt d\vartheta + c_\varepsilon \int _{\mathcal H} \varphi ^{\frac{-1}{p-1}}|\varphi _t|^{\frac{p}{p-1}}\,dt d\vartheta , \end{aligned}$$

(2.3)

for some positive constant $c_\varepsilon$.

Similarly, we have

$$\begin{aligned} \int _{\mathcal H} |u| |(\Delta _{\mathbb H}\varphi )_t|\,dtd\vartheta \le \varepsilon \int _{\mathcal H} |u|^p\varphi \,dt d\vartheta + c_\varepsilon \int _{\mathcal H}\varphi ^{\frac{-1}{p-1}} |(\Delta _{\mathbb H}\varphi )_t|^{\frac{P}{P-1}}\,dtd\vartheta \end{aligned}$$

(2.4)

and

$$\begin{aligned} \int _{\mathcal H} |u| |\Delta _{\mathbb H}\varphi |\,dtd\vartheta \le \varepsilon \int _{\mathcal H} |u|^p\varphi \,dt d\vartheta + c_\varepsilon \int _{\mathcal H}\varphi ^{\frac{-1}{p-1}} |\Delta _{\mathbb H}\varphi |^{\frac{P}{P-1}}\,dtd\vartheta . \end{aligned}$$

(2.5)

Using (2.2)–(2.5), for $\varepsilon >0$ small enough, we get

$$\begin{aligned}&\int _{\mathcal H} |u|^p\varphi \,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi (0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi \,dtd\vartheta \nonumber \\&\quad \le C \left( A_p(\varphi )+B_p(\varphi )+C_p(\varphi )+ \int _{\mathbb {H}}|u_0(\vartheta )| |\Delta _{\mathbb H}\varphi (0,\vartheta )|\,d\vartheta \right) , \end{aligned}$$

(2.6)

where

$$\begin{aligned} A_p(\varphi )= & {} \int _{\mathcal H} \varphi ^{\frac{-1}{p-1}}|\varphi _t|^{\frac{p}{p-1}}\,dt d\vartheta ,\end{aligned}$$

(2.7)

$$\begin{aligned} B_p(\varphi )= & {} \int _{\mathcal H} \varphi ^{\frac{-1}{p-1}}|(\Delta _{\mathbb H}\varphi )_t|^{\frac{p}{p-1}}\,dt d\vartheta ,\end{aligned}$$

(2.8)

$$\begin{aligned} C_p(\varphi )= & {} \int _{\mathcal H} \varphi ^{\frac{-1}{p-1}}|\Delta _{\mathbb H}\varphi |^{\frac{p}{p-1}}\,dt d\vartheta . \end{aligned}$$

(2.9)

Now, let us consider the test function

$$\begin{aligned} \varphi _R(t,\vartheta )=\phi ^\omega \left( \frac{t^2+|x|^4+|y|^4+\tau ^2}{R^4}\right) ,\quad \,\, R>0, \,\,\omega \gg 1, \end{aligned}$$

(2.10)

where $\phi \in C_0^\infty (\mathbb {R}^+)$ is a decreasing function satisfying

$$\begin{aligned} \phi (r)=\left\{ \begin{array}{lll} 1 &{}\quad \text{ if } &{} 0\le r\le 1,\\ 0 &{}\quad \text{ if } &{} r\ge 2. \end{array} \right. \end{aligned}$$

Observe that $\text{ supp }(\varphi _R)$ is a subset of

$$\begin{aligned} \Omega _R=\{(t,x,y,\tau )\in \mathcal {H}:\,0\le t^2+|x|^4+|y|^4+\tau ^2\le 2R^4\}, \end{aligned}$$

while $\text{ supp }({\varphi _R}_t)$, $\text{ supp }(\Delta _\mathbb {H}\varphi _R)$ and $\text{ supp }((\Delta _{\mathbb H}\varphi _R)_{t})$ are subsets of

$$\begin{aligned} \Theta _R=\{(t,x,y,\tau )\in \mathcal H:\, R^4\le t^2+|x|^4+|y|^4+\tau ^2\le 2R^4\}. \end{aligned}$$

Let

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

Then we have

$$\begin{aligned} \Delta _{\mathbb H} \varphi _R(t,\vartheta )= & {} \frac{4\omega (N+4)}{R^4} (|x|^{2}+|y|^{2})\phi '(\rho )\phi ^{\omega -1}(\rho )\\&+\,\frac{16\omega (\omega -1)}{R^8} ((|x|^6+|y|^6)+2\tau (|x|^2-|y|^{2})x\cdot y+\tau ^{2}(|x|^2+|y|^2))\\&\times \, \phi '^2(\rho )\phi ^{\omega -2}(\rho )\\&+\,\frac{16\omega }{R^{8}} ((|x|^{6}+|y|^{6})+2\tau (|x|^{2}-|y|^{2})x\cdot y+\tau ^{2}(|x|^2+|y|^2))\\&\times \, \phi ''(\rho )\phi ^{\omega -1}(\rho ) \end{aligned}$$

and

$$\begin{aligned} (\Delta _{\mathbb H} \varphi _R)_t(t,\vartheta )= & {} \frac{8\omega (N+4)t}{R^8} (|x|^{2}+|y|^{2})(\phi ''(\rho )\phi (\rho )+(\omega -1)\phi '^2(\rho )) \phi ^{\omega -2}(\rho )\\&+\,\frac{32\omega (\omega -1)t}{R^{12}} ((|x|^6+|y|^6)+2\tau (|x|^2-|y|^{2})x\cdot y+\tau ^{2}(|x|^2+|y|^2))\\&\times \, (2\phi (\rho )\phi '(\rho )\phi ''(\rho )+(\omega -2)\phi '^3(\rho ))\phi ^{\omega -3}(\rho )\\&+\,\frac{32\omega t}{R^{12}} ((|x|^{6}+|y|^{6})+2\tau (|x|^{2}-|y|^{2})x\cdot y+\tau ^{2}(|x|^2+|y|^2))\\&\times \, (\phi (\rho )\phi '''(\rho )+(\omega -1)\phi '(\rho )\phi ''(\rho ))\phi ^{\omega -2}(\rho ). \end{aligned}$$

It follows that there is a positive constant $C>0$, independent of R, such that for all $(t,\vartheta )\in \Omega _R$, we have

$$\begin{aligned} |\Delta _{\mathbb H} \varphi _R(t,\vartheta )|\le C R^{-2}\phi ^{\omega -2}(\rho )\chi (\rho ), \end{aligned}$$

(2.11)

where

$$\begin{aligned} \chi (\rho )=|\phi '(\rho )|\phi (\rho )+\phi '^2(\rho )+|\phi ''(\rho )|\phi (\rho ), \end{aligned}$$

and

$$\begin{aligned} |(\Delta _{\mathbb H} \varphi _R)_t(t,\vartheta )|\le C R^{-4}\phi ^{\omega -3}(\rho )\xi (\rho ), \end{aligned}$$

(2.12)

where

$$\begin{aligned} \xi (\rho )= & {} \phi ^2(\rho )|\phi ''(\rho )|+\phi '^2(\rho )\phi (\rho )+\phi (\rho )|\phi '(\rho )||\phi ''(\rho )|+|\phi '^3(\rho )|\\&+\,\phi ^2(\rho )|\phi '''(\rho )|. \end{aligned}$$

Using (2.11) and (2.12), we get

$$\begin{aligned} B_p(\varphi _R)\le & {} C R^{\frac{-4p}{p-1}}\int _{\mathcal H} \phi ^{\omega -\frac{3p}{p-1}}(\rho )\xi ^{\frac{p}{p-1}}(\rho )\,dtd\vartheta ,\end{aligned}$$

(2.13)

$$\begin{aligned} C_p(\varphi _R)\le & {} C R^{\frac{-2p}{p-1}}\int _{\mathcal H} \phi ^{\omega -\frac{2p}{p-1}}(\rho )\chi ^{\frac{p}{p-1}}(\rho )\,dtd\vartheta . \end{aligned}$$

(2.14)

Let us consider now the change of variables

$$\begin{aligned} (t,x,y,\tau )=(t,\vartheta )\mapsto (\widetilde{t},\widetilde{v})=(\widetilde{t},\widetilde{x},\widetilde{y},\widetilde{\tau }), \end{aligned}$$

(2.15)

where

$$\begin{aligned} \widetilde{t}=R^{-2}t,\quad \,\widetilde{x}=R^{-1}x,\, \widetilde{y}=R^{-1}y,\quad \, \widetilde{\tau }=R^{-2}\tau . \end{aligned}$$

Let

$$\begin{aligned} \widetilde{\rho }= & {} \widetilde{t}^2+|\widetilde{x}|^{4}+|\widetilde{y}|^4+\widetilde{\tau }^2,\\ \widetilde{\Theta }= & {} \{(\widetilde{t},\widetilde{x},\widetilde{y},\widetilde{\tau })\in \mathcal {H}:\, 1\le \widetilde{\rho }\le 2\},\\ \Sigma _R= & {} \{(x,y,\tau )\in \mathbb {H}:\, R^4\le |x|^4+|y|^4+\tau ^2\le 2R^4\}, \end{aligned}$$

and

$$\begin{aligned} C_\phi= & {} \max \left\{ \int _{\widetilde{\Theta }}\phi ^{\omega -\frac{p}{p-1}}(\widetilde{\rho })|\phi _{\widetilde{t}}^{\frac{p}{p-1}}(\widetilde{\rho })|\,d\widetilde{t}d\widetilde{\vartheta }, \int _{\widetilde{\Theta }}\phi ^{\omega -\frac{3p}{p-1}}(\widetilde{\rho })\xi ^{\frac{p}{p-1}}(\widetilde{\rho })\,d\widetilde{t}d\widetilde{\vartheta },\right. \\&\left. \times \int _{\widetilde{\Theta }}\phi ^{\omega -\frac{2p}{p-1}}(\widetilde{\rho })\chi ^{\frac{p}{p-1}}(\widetilde{\rho })\,d\widetilde{t}d\widetilde{\vartheta }\right\} . \end{aligned}$$

Using (2.6), (2.13) and (2.14), we obtain

$$\begin{aligned}&\int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi _R\,dtd\vartheta \nonumber \\&\quad \le CC_\phi \left( R^{\lambda _1}+R^{\lambda _2}+\int _{\Sigma _R}|u_0(\vartheta )| |\Delta _{\mathbb H}\varphi _R(0,\vartheta )|\,d\vartheta \right) , \end{aligned}$$

(2.16)

where

$$\begin{aligned} \lambda _1=Q+2-\frac{2p}{p-1}\quad \text{ and }\quad \lambda _2=Q+2-\frac{4p}{p-1}. \end{aligned}$$

On the other hand, we have

$$\begin{aligned}&\liminf _{R\rightarrow \infty } \left( \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi _R\,dtd\vartheta \right) \\&\quad \ge \liminf _{R\rightarrow \infty }\int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta +\liminf _{R\rightarrow \infty }\int _{\mathbb {H}}u_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\liminf _{R\rightarrow \infty }\int _{\mathcal H}f\varphi _R\,dtd\vartheta . \end{aligned}$$

Using the monotone convergence theorem, we get

$$\begin{aligned} \liminf _{R\rightarrow \infty }\int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta =\int _{\mathcal H} |u|^p\,dt d\vartheta . \end{aligned}$$

Since $u_0\in L^1(\mathbb {H})$, by the dominated convergence theorem, we have

$$\begin{aligned} \liminf _{R\rightarrow \infty }\int _{\mathbb {H}}u_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta = \int _{\mathbb {H}}u_0(\vartheta )\,d\vartheta . \end{aligned}$$

Writing $f=f^+-f^-$, where $f^+=\max \{f,0\}$, we have

$$\begin{aligned} \int _{\mathcal H}f\varphi _R\,dtd\vartheta&=\int _{\mathcal {U}_R}f\,dtd\vartheta +\int _{\Theta _R}f^+\varphi _R\,dtd\vartheta -\int _{\Theta _R}f^-\varphi _R\,dtd\vartheta \\&\ge \int _{\mathcal {U}_R}f\,dtd\vartheta -\int _{\Theta _R}f^-\varphi _R\,dtd\vartheta . \end{aligned}$$

Since $f^-\in L^1(\mathcal H)$, by the dominated convergence theorem we have

$$\begin{aligned} \lim _{R\rightarrow \infty }\int _{\Theta _R}f^-\varphi _R\,dtd\vartheta =0. \end{aligned}$$

Then

$$\begin{aligned} \liminf _{R\rightarrow \infty }\int _{\mathcal H}f\varphi _R\,dtd\vartheta \ge \liminf _{R\rightarrow \infty }\int _{\mathcal {U}_R}f\,dtd\vartheta . \end{aligned}$$

Now, we have

$$\begin{aligned}&\liminf _{R\rightarrow \infty } \left( \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi _R\,dtd\vartheta \right) \\&\quad \ge \int _{\mathcal H} |u|^p\,dt d\vartheta +\ell , \end{aligned}$$

where form (2.1),

$$\begin{aligned} \ell =\int _{\mathbb {H}}u_0(\vartheta )\,d\vartheta +\liminf _{R\rightarrow \infty }\int _{\mathcal {U}_R} f\,dtd\vartheta >0. \end{aligned}$$

By the definition of the limit inferior, for every $\varepsilon >0$, there exists $R_0>0$ such that

$$\begin{aligned}&\int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi _R\,dtd\vartheta \\&\quad > \liminf _{R\rightarrow \infty } \left( \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi _R\,dtd\vartheta \right) -\varepsilon \\&\quad \ge \int _{\mathcal H} |u|^p\,dt d\vartheta +\ell -\varepsilon , \end{aligned}$$

for every $R\ge R_0$. Taking $\varepsilon =\ell /2$, we obtain

$$\begin{aligned}&\int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi _R\,dtd\vartheta \\&\quad \ge \int _{\mathcal H} |u|^p\,dt d\vartheta +\frac{\ell }{2}, \end{aligned}$$

for every $R\ge R_0$. Then from (2.16), we have

$$\begin{aligned} \int _{\mathcal H} |u|^p\,dt d\vartheta +\frac{\ell }{2}\le CC_\phi \left( R^{\lambda _1}+R^{\lambda _2}+\int _{\Sigma _R}|u_0(\vartheta )| |\Delta _{\mathbb H}\varphi _R(0,\vartheta )|\,d\vartheta \right) , \end{aligned}$$

(2.17)

for R large enough.

Now, we require that $\lambda _1=\max \{\lambda _1,\lambda _2\}\le 0$, which is equivalent to $1<p\le 1+\displaystyle \frac{2}{Q}$. We distinguish two cases.

• Case 1. If $1<p< 1+\displaystyle \frac{2}{Q}$.

  In this case, letting $R\rightarrow \infty$ in (2.17) and using the dominated convergence theorem, we obtain

  $$\begin{aligned} \int _{\mathcal H} |u|^p\,dt d\vartheta +\frac{\ell }{2}\le 0, \end{aligned}$$

  which is a contradiction with $\ell >0$.

• Case 2. If $p=1+\displaystyle \frac{2}{Q}$.

  In this case, from (2.17), we obtain

  $$\begin{aligned} \int _{\mathcal H} |u|^p\,dt d\vartheta \le C<\infty . \end{aligned}$$

  (2.18)

  Using the Hölder inequality with parameters p and $p/(p-1)$, from (2.2), we obtain

  $$\begin{aligned} \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta +\frac{\ell }{2}\le C \left( \int _{\Theta _R} |u|^p\varphi _R\,dt d\vartheta \right) ^{\frac{1}{p}}. \end{aligned}$$

  Letting $R\rightarrow \infty$ in the above inequality and using (2.18), we obtain

  $$\begin{aligned} \int _{\mathcal H} |u|^p\,dt d\vartheta +\frac{\ell }{2}=0. \end{aligned}$$

  This contradiction completes the proof of the theorem.

$\square$

2.2 The case of $2\times 2$ systems

Let $f,g\in L^1_{loc}(\mathcal H)$.

2.2.1 The case of system (1.3)

The definition of solutions we adopt for (1.3) is:

Definition 2.3

We say that the pair (u, v) is a global weak solution to (1.3) on $\mathcal {H}$ with initial data $(u(0,\cdot ),v(0,\cdot ))=(u_0,v_0)\in L^1_{loc}(\mathbb {H})\times L^1_{loc}(\mathbb {H})$, if $(u,v)\in L^{p}_{loc}(\mathcal H)\times L^{q}_{loc}(\mathcal H)$ and satisfies

$$\begin{aligned}&\int _{\mathcal H} |v|^q\varphi \,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi (0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi \,dtd\vartheta \\&\quad =-\int _{\mathcal H} u\varphi _t\,dt d\vartheta +\int _{\mathcal H} u (\Delta _{\mathbb H}\varphi )_t\,dtd\vartheta -\int _{\mathcal H}u \Delta _{\mathbb {H}}\varphi \,dtd\vartheta \\&\qquad +\,\int _{\mathbb {H}}u_0(\vartheta )\Delta _{\mathbb H}\varphi (0,\vartheta )\,d\vartheta \end{aligned}$$

and

$$\begin{aligned}&\int _{\mathcal H} |u|^p\varphi \,dt d\vartheta +\int _{\mathbb {H}}v_0(\vartheta )\varphi (0,\vartheta )\,d\vartheta +\int _{\mathcal H}g\varphi \,dtd\vartheta \\&\quad =-\int _{\mathcal H} v\varphi _t\,dt d\vartheta +\int _{\mathcal H} v (\Delta _{\mathbb H}\varphi )_t\,dtd\vartheta -\int _{\mathcal H}v \Delta _{\mathbb {H}}\varphi \,dtd\vartheta \\&\qquad +\,\int _{\mathbb {H}}v_0(\vartheta )\Delta _{\mathbb H}\varphi (0,\vartheta )\,d\vartheta , \end{aligned}$$

for any regular test function $\varphi$, $\varphi (\cdot ,t)=0$, $t\ge T$.

Our second main result is given by the following theorem.

Theorem 2.4

Let $(u_0,v_0)\in L^1(\mathbb {H})\times L^1(\mathbb {H})$ and $(f^-,g^-)\in L^1(\mathcal H)\times L^1(\mathcal H)$. Suppose that

$$\begin{aligned} \int _{\mathbb {H}}u_0\,d\vartheta +\liminf _{R\rightarrow \infty }\int _{\mathcal {U}_R} f\,dtd\vartheta > 0 \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {H}}v_0\,d\vartheta +\liminf _{R\rightarrow \infty }\int _{\mathcal {U}_R} g\,dtd\vartheta > 0. \end{aligned}$$

If $1<pq\le (pq)^*$, where

$$\begin{aligned} (pq)^*=1+\frac{2}{Q}\max \{p+1,q+1\}, \end{aligned}$$

then there exists no nontrivial global weak solution to (1.3).

Proof

Suppose that (u, v) is a nontrivial global weak solution to (1.3). Then for any regular test function $\varphi$, we have

$$\begin{aligned}&\int _{\mathcal H} |v|^q\varphi \,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi (0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi \,dtd\vartheta \\&\quad \le \int _{\mathcal H} |u||\varphi _t|\,dt d\vartheta +\int _{\mathcal H} |u| |(\Delta _{\mathbb H}\varphi )_t|\,dtd\vartheta +\int _{\mathcal H}|u| |\Delta _{\mathbb {H}}\varphi |\,dtd\vartheta \\&\qquad +\,\int _{\mathbb {H}}|u_0(\vartheta )||\Delta _{\mathbb H}\varphi (0,\vartheta )|\,d\vartheta \end{aligned}$$

and

$$\begin{aligned}&\int _{\mathcal H} |u|^p\varphi \,dt d\vartheta +\int _{\mathbb {H}}v_0(\vartheta )\varphi (0,\vartheta )\,d\vartheta +\int _{\mathcal H}g\varphi \,dtd\vartheta \\&\quad \le \int _{\mathcal H} |v||\varphi _t|\,dt d\vartheta +\int _{\mathcal H} |v| |(\Delta _{\mathbb H}\varphi )_t|\,dtd\vartheta +\int _{\mathcal H}|v| |\Delta _{\mathbb {H}}\varphi |\,dtd\vartheta \\&\qquad +\,\int _{\mathbb {H}}|v_0(\vartheta )||\Delta _{\mathbb H}\varphi (0,\vartheta )|\,d\vartheta . \end{aligned}$$

Taking $\varphi =\varphi _R$, the test function given by (2.10), using the Hölder inequality with parameters p and $p/(p-1)$, we get

$$\begin{aligned}&\int _{\mathcal H} |v|^q\varphi _R\,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi _R\,dtd\vartheta \\&\qquad -\int _{\mathbb {H}}|u_0(\vartheta )||\Delta _{\mathbb H}\varphi _R(0,\vartheta )|\,d\vartheta \\&\quad \le \left( A_p(\varphi _R)^{\frac{p-1}{p}}+B_p(\varphi _R)^{\frac{p-1}{p}}+C_p(\varphi _R)^{\frac{p-1}{p}}\right) \left( \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta \right) ^{\frac{1}{p}}, \end{aligned}$$

where $A_p(\varphi )$, $B_p(\varphi )$ and $C_p(\varphi )$ are given respectively by (2.7)–(2.9). Similarly, by the Hölder inequality with parameters q and $q/(q-1)$, we get

$$\begin{aligned}&\int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta +\int _{\mathbb {H}}v_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\int _{\mathcal H}g\varphi _R\,dtd\vartheta \\&\qquad -\,\int _{\mathbb {H}}|v_0(\vartheta )||\Delta _{\mathbb H}\varphi _R(0,\vartheta )|\,d\vartheta \\&\quad \le \left( A_q(\varphi _R)^{\frac{q-1}{q}}+B_q(\varphi _R)^{\frac{q-1}{q}}+C_q(\varphi _R)^{\frac{q-1}{q}}\right) \left( \int _{\mathcal H} |v|^q\varphi _R\,dt d\vartheta \right) ^{\frac{1}{q}}. \end{aligned}$$

Without restriction of the generality, we may assume that for R large enough, we have

$$\begin{aligned} \int _{\mathbb {H}}u_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi _R\,dtd\vartheta -\int _{\mathbb {H}}|u_0(\vartheta )||\Delta _{\mathbb H}\varphi _R(0,\vartheta )|\,d\vartheta \ge 0 \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {H}}v_0(\vartheta )\varphi _R(0,\vartheta )\,d\vartheta +\int _{\mathcal H}g\varphi _R\,dtd\vartheta -\int _{\mathbb {H}}|v_0(\vartheta )||\Delta _{\mathbb H}\varphi _R(0,\vartheta )|\,d\vartheta \ge 0. \end{aligned}$$

Slight modifications yield the proof in the general case (see the proof of Theorem 2.2). Then for R large enough, we have

$$\begin{aligned} \int _{\mathcal H} |v|^q\varphi _R\,dt d\vartheta \le (A_p(\varphi _R)^{\frac{p-1}{p}}+B_p(\varphi _R)^{\frac{p-1}{p}}+C_p(\varphi _R)^{\frac{p-1}{p}}) \left( \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta \right) ^{\frac{1}{p}} \end{aligned}$$

(2.19)

and

$$\begin{aligned} \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta \le (A_q(\varphi _R)^{\frac{q-1}{q}}+B_q(\varphi _R)^{\frac{q-1}{q}}+C_q(\varphi _R)^{\frac{q-1}{q}}) \left( \int _{\mathcal H} |v|^q\varphi _R\,dt d\vartheta \right) ^{\frac{1}{q}}. \end{aligned}$$

(2.20)

Using the change of variables (2.15), from (2.19) and (2.20), we obtain

$$\begin{aligned} \int _{\mathcal H} |v|^q\varphi _R\,dt d\vartheta \le CR^{\frac{Q(p-1)-2}{p}} \left( \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta \right) ^{\frac{1}{p}} \end{aligned}$$

(2.21)

and

$$\begin{aligned} \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta \le CR^{\frac{Q(q-1)-2}{q}}\left( \int _{\mathcal H} |v|^q\varphi _R\,dt d\vartheta \right) ^{\frac{1}{q}}. \end{aligned}$$

(2.22)

Combining (2.21) with (2.22), we obtain

$$\begin{aligned} \left( \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta \right) ^{1-\frac{1}{pq}} \le CR^{\upsilon _1} \end{aligned}$$

(2.23)

and

$$\begin{aligned} \left( \int _{\mathcal H} |v|^q\varphi _R\,dt d\vartheta \right) ^{1-\frac{1}{pq}} \le CR^{\upsilon _2}, \end{aligned}$$

(2.24)

where

$$\begin{aligned} \upsilon _1=\frac{Q(pq-1)-2(p+1)}{pq}\quad \text{ and }\quad \upsilon _2=\frac{Q(pq-1)-2(q+1)}{pq}. \end{aligned}$$

We require that $\upsilon _1\le 0$ or $\upsilon _2\le 0$ which is equivalent to $1<pq\le 1+\frac{2}{Q}\max \{p+1,q+1\}$. We distinguish two case.

• Case 1. If $1<pq< 1+\frac{2}{Q}\max \{p+1,q+1\}$.

  Without restriction of the generality, we may suppose that $0<q\le p$. In this case, letting $R\rightarrow \infty$ in (2.23), we obtain

  $$\begin{aligned} \int _{\mathcal H} |u|^p\,dt d\vartheta =0, \end{aligned}$$

  which is a contradiction.

• Case 2. If $pq=1+\frac{2}{Q}\max \{p+1,q+1\}$.

  This case can be treated in the same way as in the proof of Theorem 2.2.

Remark 2.5

If $p=q$ and $u=v$ in Theorem 2.4, we obtain the result for a single equation given by Theorem 2.2.

2.2.2 The case of system (1.4)

The definition of solutions we adopt for (1.4) is:

Definition 2.6

We say that the pair (u, v) is a global weak solution to (1.3) on $\mathcal {H}$ with initial data $(u(0,\cdot ),v(0,\cdot ))=(u_0,v_0)\in L^1_{loc}(\mathbb {H})\times L^1_{loc}(\mathbb {H})$, if $(u,v)\in L^{p}_{loc}(\mathcal H)\times L^{q}_{loc}(\mathcal H)$ and satisfies

$$\begin{aligned}&\int _{\mathcal H} |v|^q\varphi \,dt d\vartheta +\int _{\mathbb {H}}u_0(\vartheta )\varphi (0,\vartheta )\,d\vartheta +\int _{\mathcal H}f\varphi \,dtd\vartheta \\&\quad =-\int _{\mathcal H} u\varphi _t\,dt d\vartheta +\int _{\mathcal H} u (\Delta _{\mathbb H}\varphi )_t\,dtd\vartheta -\int _{\mathcal H}v \Delta _{\mathbb {H}}\varphi \,dtd\vartheta \\&\qquad +\,\int _{\mathbb {H}}u_0(\vartheta )\Delta _{\mathbb H}\varphi (0,\vartheta )\,d\vartheta \end{aligned}$$

and

$$\begin{aligned}&\int _{\mathcal H} |u|^p\varphi \,dt d\vartheta +\int _{\mathbb {H}}v_0(\vartheta )\varphi (0,\vartheta )\,d\vartheta +\int _{\mathcal H}g\varphi \,dtd\vartheta \\&\quad =-\int _{\mathcal H} v\varphi _t\,dt d\vartheta +\int _{\mathcal H} v (\Delta _{\mathbb H}\varphi )_t\,dtd\vartheta -\int _{\mathcal H}u \Delta _{\mathbb {H}}\varphi \,dtd\vartheta \\&\qquad +\,\int _{\mathbb {H}}v_0(\vartheta )\Delta _{\mathbb H}\varphi (0,\vartheta )\,d\vartheta , \end{aligned}$$

for any regular test function $\varphi$, $\varphi (\cdot ,t)=0$, $t\ge T$.

We have the following result.

Theorem 2.7

Let $(u_0,v_0)\in L^1(\mathbb {H})\times L^1(\mathbb {H})$ and $(f^-,g^-)\in L^1(\mathcal H)\times L^1(\mathcal H)$. Suppose that

$$\begin{aligned} \int _{\mathbb {H}}u_0\,d\vartheta +\liminf _{R\rightarrow \infty }\int _{\mathcal {U}_R} f\,dtd\vartheta > 0 \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {H}}v_0\,d\vartheta +\liminf _{R\rightarrow \infty }\int _{\mathcal {U}_R} g\,dtd\vartheta > 0. \end{aligned}$$

If

$$\begin{aligned} Q\le 2\max \{Q_1,Q_2\}, \end{aligned}$$

(2.25)

where

$$\begin{aligned} Q_1= & {} \min \left\{ \frac{1}{p-1},\frac{1}{q-1}\left( 1+\frac{q^2}{q+1}\right) ,\frac{2}{q+1}\left( \frac{q^2}{q-1}+\frac{p}{p-1}\right) -1\right\} ,\\ Q_2= & {} \min \left\{ \frac{1}{q-1},\frac{1}{p-1}\left( 1+\frac{p^2}{p+1}\right) ,\frac{2}{p+1}\left( \frac{p^2}{p-1}+\frac{q}{q-1}\right) -1\right\} , \end{aligned}$$

then there exists no nontrivial global weak solution to (1.4).

Proof

Suppose that (u, v) is a nontrivial weak solution to (1.4). We continue to use the same notations of the proof of Theorem 2.4. By proceeding in the same manner as in the proof of Theorem 2.4, for R large enough, we obtain

$$\begin{aligned} X^p\le C_p(\varphi _R)^{\frac{p-1}{p}} X+D_q(\varphi _R) Y \end{aligned}$$

and

$$\begin{aligned} Y^q\le D_p(\varphi _R) X+C_q(\varphi _R)^{\frac{q-1}{q}} Y, \end{aligned}$$

where

$$\begin{aligned}&X=\left( \int _{\mathcal H} |u|^p\varphi _R\,dt d\vartheta \right) ^{\frac{1}{p}},\quad Y=\left( \int _{\mathcal H} |v|^q\varphi _R\,dt d\vartheta \right) ^{\frac{1}{q}},\\&D_p(\varphi _R)= A_p(\varphi _R)^{\frac{p-1}{p}}+ B_p(\varphi _R)^{\frac{p-1}{p}}. \end{aligned}$$

Using Lemma 3 in [11], we obtain

$$\begin{aligned} X^{pq}\le C\left( C_p(\varphi _R)^q+C_q(\varphi _R)D_q(\varphi _R)^q+(D_q(\varphi _R)^qD_p(\varphi _R))^{\frac{pq}{pq-1}}\right) \end{aligned}$$

and

$$\begin{aligned} Y^{pq}\le C\left( C_q(\varphi _R)^p+C_p(\varphi _R)D_p(\varphi _R)^p+(D_p(\varphi _R)^pD_q(\varphi _R))^{\frac{pq}{pq-1}}\right) . \end{aligned}$$

Using the change of variables (2.15), we get

$$\begin{aligned} X^{pq}\le C(R^{\rho _1}+R^{\rho _2}+R^{\rho _3}), \end{aligned}$$

where

$$\begin{aligned} \rho _1= & {} q\left( Q+2-\frac{2p}{p-1}\right) ,\\ \rho _2= & {} (Q+2)(q+1)-\frac{2q}{q-1}(2q+1),\\ \rho _3= & {} \frac{pq}{pq-1}\left( (Q+2)(q+1)-\frac{4p}{p-1}-\frac{4q^2}{q-1}\right) , \end{aligned}$$

and

$$\begin{aligned} Y^{pq}\le C(R^{\nu _1}+R^{\nu _2}+R^{\nu _3}), \end{aligned}$$

where

$$\begin{aligned} \nu _1= & {} p\left( Q+2-\frac{2q}{q-1}\right) ,\\ \nu _2= & {} (Q+2)(p+1)-\frac{2p}{p-1}(2p+1),\\ \nu _3= & {} \frac{pq}{pq-1}\left( (Q+2)(p+1)-\frac{4q}{q-1}-\frac{4p^2}{p-1}\right) . \end{aligned}$$

As in the previous proof, to get a contradiction, we have just to take $\max \{\rho _1,\rho _2,\rho _3\}\le 0$ or $\max \{\nu _1,\nu _2,\nu _3\}\le 0$, which is equivalent to (2.25). This ends the proof of Theorem 2.7. $\square$

Remark 2.8

If $p=q$ and $u=v$ in Theorem 2.7, we obtain the result for a single equation given by Theorem 2.2.