Abstract
We derive sufficient conditions for the nonexistence of global weak solutions to the nonlinear pseudo-parabolic equation
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.
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1 Introduction
In this paper, we are concerned with nonexistence of global weak solutions to the nonlinear pseudo-parabolic equation
under the initial condition
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
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
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
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
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
for \(i=1,\ldots ,N\), as follows
that is,
For all \(\vartheta ,\vartheta '\in \mathbb {H}\), we have
For \(\lambda \in \mathbb {R}\) and \((x,y,\tau )\in \mathbb {H}\), we have
If \(u(\vartheta )=v(|\vartheta |_\mathbb {H})\), then
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
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
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
If
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
Using the \(\varepsilon \)-Young inequality with parameters p and \(p/(p-1)\), we obtain
for some positive constant \(c_\varepsilon \).
Similarly, we have
and
Using (2.2)–(2.5), for \(\varepsilon >0\) small enough, we get
where
Now, let us consider the test function
where \(\phi \in C_0^\infty (\mathbb {R}^+)\) is a decreasing function satisfying
Observe that \(\text{ supp }(\varphi _R)\) is a subset of
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
Let
Then we have
and
It follows that there is a positive constant \(C>0\), independent of R, such that for all \((t,\vartheta )\in \Omega _R\), we have
where
and
where
Using (2.11) and (2.12), we get
Let us consider now the change of variables
where
Let
and
Using (2.6), (2.13) and (2.14), we obtain
where
On the other hand, we have
Using the monotone convergence theorem, we get
Since \(u_0\in L^1(\mathbb {H})\), by the dominated convergence theorem, we have
Writing \(f=f^+-f^-\), where \(f^+=\max \{f,0\}\), we have
Since \(f^-\in L^1(\mathcal H)\), by the dominated convergence theorem we have
Then
Now, we have
where form (2.1),
By the definition of the limit inferior, for every \(\varepsilon >0\), there exists \(R_0>0\) such that
for every \(R\ge R_0\). Taking \(\varepsilon =\ell /2\), we obtain
for every \(R\ge R_0\). Then from (2.16), we have
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
and
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
and
If \(1<pq\le (pq)^*\), where
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
and
Taking \(\varphi =\varphi _R\), the test function given by (2.10), using the Hölder inequality with parameters p and \(p/(p-1)\), we get
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
Without restriction of the generality, we may assume that for R large enough, we have
and
Slight modifications yield the proof in the general case (see the proof of Theorem 2.2). Then for R large enough, we have
and
Using the change of variables (2.15), from (2.19) and (2.20), we obtain
and
Combining (2.21) with (2.22), we obtain
and
where
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
and
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
and
If
where
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
and
where
Using Lemma 3 in [11], we obtain
and
Using the change of variables (2.15), we get
where
and
where
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.
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Acknowledgments
M. Jleli and B. Samet would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project No RGP-1435-034.
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Jleli, M., Kirane, M. & Samet, B. Nonexistence results for pseudo-parabolic equations in the Heisenberg group. Monatsh Math 180, 255–270 (2016). https://doi.org/10.1007/s00605-015-0823-7
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DOI: https://doi.org/10.1007/s00605-015-0823-7