# Comptes Rendus
## Mécanique

Jun Zhou and Xiongrui Wang

Blow-up time of solutions to a class of pseudo-parabolic equations

Volume 351 (2023), p. 219-226

Published online: 25 May 2023

https://doi.org/10.5802/crmeca.189

<figure><img src="image_1.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/

<figure><img src="image_2.png" /></figure>

Les Comptes Rendus. Mécanique sont membres du
Centre Mersenne pour l'édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1873-7234
<figure><img src="image_3.png" /></figure>

Spontaneous Articles / Articles spontanés

Blow-up time of solutions to a class of
pseudo-parabolic equations

Jun Zhou*, a and Xiongrui Wangb

**Abstract.** In this paper, we study the Dirichlet problem for a semilinear pseudo-parabolic equation. By using the energy estimates and ordinary differential inequalities, we studied the upper and lower bounds of blow-up time of the solutions. The results of this paper extend and complete the results on this model.

**Keywords.** Pseudo-parabolic equation, Blow-up, Blow-up time, Potential well method, Global existence.

**Mathematical subject classification (2010).** 35K70, 35A01.

**Funding.** This work is supported by Natural Science Foundation of Chongqing (CSTB2022NSCQ-MSX1674).
*Manuscript received 25 October 2022, revised and accepted 7 April 2023.*

# 1. Introduction

The pseudo-parabolic equation

$$u_t - a\Delta u_t - \Delta u = f(u) \quad (1)$$

with $a > 0$ can be used to describe many interesting physical and biological phenomena [1–6], for example, the non-stationary process in semiconductors in the presence of sources, where $a\Delta u_t - u_t$ stands for the free electron density rate, $\Delta u$ stands for the linear dissipation of free charge current and the nonlinear term stands for the source of free electron current (see [3, 5]). Showalter and Ting [7] investigated the initial boundary value problem of (1) with $f(u) = 0$, and the global existence, uniqueness and regularity of solutions were studied. When $f(u)$ is a polynomial, i.e., $f(u) = u^{p-1}$ or $f(u) = |u|^{p-2}u$, the initial and boundary value problem of (1) was studied in [8–13], and the existence, asymptotic behavior of the global solutions and global nonexistence of solutions were studied. When $f(u)$ is a logarithmic function, i.e., $f(u) = u\ln|u|$, the initial boundary value problem of (1) was studied in [14], where global existence, infinite-time blow-up of solutions, and behavior of vacuum isolation of solutions were studied. When $f(u)$ is a nonlocal function, i.e., $f(u) = |u|^{p-2}u - \frac{1}{|\Omega|}\int_{\Omega}|u|^{p-2}udx$, where $\Omega \subset \mathbb{R}^N$ is a bounded domain, problem (1) with homogeneous Neumann boundary and initial value was studied [15],

<footnote>a School of Mathematics and Statistics, Southwest University, Chongqing, 400715,
P. R. China</footnote>

<footnote>b Department of Mathematics, Yibin University, Yibin, Sichuan, 644000, P. R. China
E-mails: jzhou@swu.edu.cn, wxr888x@163.com</footnote>

<footnote>* Corresponding author.</footnote>
and the existence, uniqueness and asymptotic behavior of the global solution and the blow-up phenomena of solution were studied. For the study of global existence and finite time blow-up of solutions for pseudo-parabolic equation with potential terms, we refer the readers to [16, 17].

In this paper, we consider the following initial boundary-value problem (IBVP) for a class of pseudo-parabolic equation proposed by Zhu el al. in [18]:

$$ egin{cases} u_t - \Delta u_t - \Delta u + u = |u|^{p-2}u, & x \in \Omega, t > 0, \\ u(x, 0) = u_0(x), & x \in \Omega, \\ u(x, t) = 0, & x \in \partial\Omega, t > 0, \end{cases} \quad (2) $$

where $p \in (2, 2^*)$ and either $\Omega = \mathbb{R}^N$ ($N \ge 3$) or $\Omega \subset \mathbb{R}^N$ ($N \ge 3$) is a bounded domain with smooth boundary $\partial\Omega$, $u_0 \in H_0^1(\Omega)$. Here

$$ 2^* = \frac{2N}{N-2}. $$

When $\Omega = \mathbb{R}^N$, the boundary condition that $u(x, t) = 0$ for $x \in \partial\Omega$ and $t > 0$ is ineffective.

Problem (2) was studied by Zhu el al. in [18] via the potential well method (see, for example, [19–21]), where the existence, uniqueness, polynomial decay of the global solutions, and blow-up of solutions were studied. Zhou [22] improve the results of [18] by showing the global solutions decay exponentially. The main purpose of this paper is to complete the blow-up result given in [18] by estimating the upper and lower bounds of the blow-up time. To introduce the previous results and give the main results of the present paper, let’s firstly recall some notations used in [18]. Throughout this paper, we use $\|\cdot\|_p$ as the usual $L^p$-norm, $(\cdot, \cdot)$ and $\|\cdot\|$ as the inner product and the associated norm on $H_0^1(\Omega)$ respectively, that is

$$ (u, v) = \int_{\Omega} (\nabla u \cdot \nabla v + uv) dx \quad (3) $$

and

$$ \|u\| = \sqrt{\langle u, u \rangle}. \quad (4) $$

By [18, Theorem 1.3], when $p \in (2, 2^*)$ and $u_0 \in H_0^1(\Omega)$, IVBP (2) admits a (weak) solution

$$ u \in C^1([0, T_{\max}), H_0^1(\Omega)), \quad (5) $$

which satisfies $u(0) = u_0 \in H_0^1(\Omega)$ and

$$ (u'(t), v) + (u(t), v) = \int_{\Omega} |u(t)|^{p-2} u(t) v dx, \quad \forall v \in H_0^1(\Omega) \quad (6) $$

where $T_{\max}$ is the maximum existence time.

As in [18], the energy functional $J(u)$ can be defined through

$$ J(u) := \frac{p-2}{2p} \|u\|^2 + \frac{1}{p} I(u), \quad \forall u \in H_0^1(\Omega), \quad (7) $$

where

$$ I(u) := \|u\|^2 - \|u\|_p^p, \quad \forall u \in H_0^1(\Omega). \quad (8) $$

By (5), $J(u(t))$ and $I(u(t))$ are well-defined and are continuous differentiable with respect to $t$.

Let

$$ \mathcal{N} := \{u \in H_0^1(\Omega) \setminus \{0\} : I(u) = 0\} \quad (9) $$

denote the Nehari manifold. We also define two sets $\mathcal{N}_{\pm}$ related to $\mathcal{N}$ as follows:

$$ \begin{aligned} \mathcal{N}_{+} &:= \{u \in H_0^1(\Omega) : I(u) > 0\}, \\ \mathcal{N}_{-} &:= \{u \in H_0^1(\Omega) : I(u) < 0\}. \end{aligned} \quad (10) $$
The main result on global existence and blow-up of [18] is the following theorem (see [18, Theorem 1.4]):

**Theorem 1.** Let $u = u(t)$ be a solution of IBVP (2), whose maximal existence time is $T_{\max}$. Then $T_{\max} < \infty$ if and only if there exists a $t_0 \in [0, T_{\max})$ such that $u(t_0) \in \mathcal{N}_{-}$. Moreover, if $T_{\max} < \infty$, then $\lim_{t \uparrow T_{\max}} \|u(t)\| = \infty$.

When blow-up occurs, the blow-up time $T_{\max}$ cannot usually be computed exactly. It is therefore of great importance in practice to determine the upper and lower bounds for $T_{\max}$. So we complete Theorem 1 by showing the upper and lower bounds of $T_{\max}$:

**Theorem 2.** Let $u = u(t)$ be a solution of IBVP (2) with initial value $u_0 \in \mathcal{N}_{-}$, i.e., $I(u_0) = \|u_0\|^2 - \|u_0\|_p^p < 0$. Then

$$ \frac{1}{2} \int_{\|u_0\|^2}^{\infty} \frac{d\xi}{\kappa^p \xi^{p/2} - \xi} \le T_{\max} \le \frac{4(p-1)\|u_0\|^2}{(p-2)^2 (\|u_0\|_p^p - \|u_0\|^2)}, \quad (11) $$

where $\kappa$ is the optimal constant of the embedding $H_0^1(\Omega) \hookrightarrow L^p(\Omega)$, i.e.,

$$ \kappa = \sup_{\substack{u \in H_0^1(\Omega) \\ \|u\|=1}} \|u\|_p. \quad (12) $$

**Remark 3.**

(1) Since $p \in (2, 2^*)$, it follows $H_0^1(\Omega) \hookrightarrow L^p(\Omega)$;

(2) We show that

$$ \frac{1}{2} \int_{\|u_0\|^2}^{\infty} \frac{d\xi}{\kappa^p \xi^{p/2} - \xi} < \frac{4(p-1)\|u_0\|^2}{(p-2)^2 (\|u_0\|_p^p - \|u_0\|^2)}. \quad (13) $$

So, (11) makes sense. By using $I(u_0) < 0$, it follows from (8) and (12) that

$$ \|u_0\|^2 < \|u_0\|_p^p \le \kappa^p \|u_0\|^p, $$

which implies

$$ \|u_0\| > \kappa^{\frac{p}{p-2}}. \quad (14) $$

Let $f(\xi) = \sigma \kappa^p \xi^{p/2} - \xi$ for $\xi \ge 0$, where

$$ \sigma = \kappa^{-p} \|u_0\|^{2-p}. $$

Since $p > 2$, by (14),

$$ \sigma < \kappa^{-p} \left( \kappa^{\frac{p}{p-2}} \right)^{2-p} = 1. $$

Then, for $\xi \in [\|u_0\|^2, \infty)$,

$$ \begin{align} f'(\xi) &= \frac{p}{2\sigma\kappa^p\xi^{p/2-1}} - 1 = \frac{p}{2\|u_0\|^{2-p}\xi^{p/2-1}} - 1 \nonumber \\ &\ge \frac{p}{2\|u_0\|^{2-p}} (\|u_0\|^{p/2-1} - 1) \tag{15} \\ &= \frac{p}{2} - 1 > 0. \nonumber \end{align} $$

So,

$$ f(\xi) \ge f(\|u_0\|^2) = 0, \xi \in [\|u_0\|^2, \infty). $$
Then, note $\sigma \in (0, 1)$, we can estimate the integral to the left of (13) as

$$ 
\begin{aligned}
\frac{1}{2} \int_{\|u_0\|^2}^{\infty} \frac{d\xi}{\kappa^p \xi^{\frac{p}{2}} - \xi} &= \frac{1}{2} \int_{\|u_0\|^2}^{\infty} \frac{d\xi}{(1-\sigma)\kappa^p \xi^{\frac{p}{2}} + f(\xi)} \\
&\le \frac{1}{2(1-\sigma)\kappa^p} \int_{\|u_0\|^2}^{\infty} \frac{d\xi}{\xi^{\frac{p}{2}}} = \frac{1}{(1-\sigma)\kappa^p(p-2)} \|u_0\|^{2-p} \quad (16) \\
&= \frac{1}{(1-\kappa^{-p}\|u_0\|^{2-p})\kappa^p(p-2)} \|u_0\|^{2-p} \\
&= \frac{1}{(p-2)(\kappa^p \|u_0\|^{p-2} - 1)}.
\end{aligned}
$$

By (12), we can estimate the right of (13) as

$$ 
\begin{aligned}
\frac{4(p-1)\|u_0\|^2}{(p-2)^2 (\|u_0\|_p^p - \|u_0\|^2)} &\ge \frac{4(p-1)\|u_0\|^2}{(p-2)^2 (\kappa^p \|u_0\|^{p-2} - \|u_0\|^2)} \\
&= \frac{4(p-1)}{(p-2)^2 (\kappa^p \|u_0\|^{p-2} - 1)}.
\end{aligned}
\quad (17) $$

Since $4(p-1)/(p-2) > 4$, (13) follows from (16) and (17).

(3) It obvious that

$$ \int_{\|u_0\|^2}^{\infty} \frac{d\xi}{\kappa^p \xi^{\frac{p}{2}} - \xi} \ge \int_{\|u_0\|^2}^{\infty} \frac{d\xi}{\kappa^p \xi^{\frac{p}{2}} - \xi} = \frac{2}{\kappa^p(p-2)} \|u_0\|^{2-p}. $$

Then it follows from (11) that

$$ \frac{1}{\kappa^p(p-2)} \|u_0\|^{2-p} \le T_{\max} \le \frac{4(p-1)\|u_0\|^2}{(p-2)^2 (\|u_0\|_p^p - \|u_0\|^2)}. $$

## 2. Proof of Theorem 2

In this section, we give the proof of Theorem 2 by using the following lemma:

**Lemma 4 ([23]).** Suppose that $0 < T \le +\infty$ and suppose a nonnegative function $F(t) \in C^2[0, T)$ satisfies

$$ F''(t)F(t) - (1+\gamma)(F'(t))^2 \ge 0 $$

for some constant $\gamma > 0$. If $F(0) > 0$, $F'(0) > 0$, then

$$ T \le \frac{F(0)}{\gamma F'(0)} < \infty $$

and $F(t) \to +\infty$ as $t \uparrow T$.

**Proof of Theorem 2.** Let $u = u(t)$ be a solution of problem (2) with initial value $u_0 \in \mathcal{N}_-$. By Theorem 1, the maximal existence $T_{\max} < \infty$. So we only need to prove the inequality (11).

(1) **Upper bound estimate.** In the part, we will give the upper bound estimate, i.e., show that

$$ T_{\max} \le \frac{4(p-1)\|u_0\|^2}{(p-2)^2 (\|u_0\|_p^p - \|u_0\|^2)}, \quad (18) $$
For $t \in [0, T_{\max})$, let

$$ \xi(t) := \left( \int_{0}^{t} \|u(s)\|^2 ds \right)^{\frac{1}{2}}, $$

$$ \eta(t) := \left( \int_{0}^{t} \|u'(s)\|^2 ds \right)^{\frac{1}{2}}. $$

The following facts can be found in [18, Lemma 3.2 and the proof of Theorem 5.1]:

$$ (R_1) \quad J(u(t)) + \eta^2(t) = J(u_0), \quad t \in [0, T_{\max}); $$

$$ (R_2) \quad \frac{d}{dt} \|u(t)\|^2 = -2I(u(t)), \quad t \in [0, T_{\max}); $$

$$ (R_3) \quad u(t) \in \mathcal{N}, \text{ i.e., } I(u(t)) < 0, t \in [0, T_{\max}). $$

By $(R_1)-(R_3)$, it is easy to see

$$ (R_4) \quad \|u(t)\|^2 \text{ is strictly increasing with respect to } t; $$

$$ (R_5) \quad \text{the function } \phi(t) := (p-2)\|u(t)\|^2 - 2p(J(u(t))) \text{ satisfies} $$

$$ \phi(t) \ge (p-2)\|u_0\|^2 - 2p(J(u_0) - \eta^2(t)) \\ = 2(\|u_0\|_p^p - \|u_0\|^2) + 2p\eta^2(t), \quad t \in [0, T_{\max}). $$ 
by (7) and (8)

Consider the following functional

$$ F(t) := \xi^2(t) + (T_{\max} - t) \|u_0\|^2 + \beta(t + \alpha)^2, \quad t \in [0, T_{\max}), \quad (19) $$

where $\alpha$ and $\beta$ are two positive constants to be determined later. Then by $(R_4)$, we have

$$ \begin{aligned} F'(t) &= \|u(t)\|^2 - \|u_0\|^2 + 2\beta(t+\alpha) \\ &\ge 2\beta(t+\alpha) > 0, \quad t \in [0, T_{\max}), \end{aligned} \quad (20) $$

which implies

$$ F(t) \ge F(0) = T_{\max} \|u_0\|^2 + \beta \alpha^2 > 0, \quad t \in [0, T_{\max}) \quad (21) $$

and (by $(R_2)$, (7) and $(R_5)$)

$$ \begin{aligned} F''(t) &= -2I((t)) + 2\beta \\ &= \phi(t) + 2\beta \\ &\ge 2(\|u_0\|_p^p - \|u_0\|^2) + 2p\eta^2(t) + 2\beta, \quad t \in [0, T_{\max}). \end{aligned} \quad (22) $$

By (3), the Cauchy–Schwarz inequality and Hölder's inequality, we have

$$ \begin{aligned} \frac{1}{2} \int_{0}^{t} \frac{d}{ds} \|u(s)\|^2 ds &= \int_{0}^{t} (u(s), u'(s)) ds \\ &\le \int_{0}^{t} \|u(s)\| \|u'(s)\| ds \\ &\le \xi(t) \eta(t), \quad t \in [0, T_{\max}), \end{aligned} $$
which, together with (19), implies

$$
\begin{aligned}
(F(t) - (T_{\max} - t) \|u_0\|_2^2) (\eta^2(t) + \beta) &= (\xi^2(t) + \beta(t+\alpha)^2) (\eta^2(t) + \beta) \\
&= \xi^2(t)\eta^2(t) + \beta\xi^2(t) + \beta(t+\alpha)^2\eta^2(t) + \beta^2(t+\alpha)^2 \\
&\geq \xi^2(t)\eta^2(t) + 2\xi(t)\eta(t)\beta(t+\alpha) + \beta^2(t+\alpha)^2 \\
&\geq (\xi(t)\eta(t) + \beta(t+\alpha))^2 \\
&\geq \left(\frac{1}{2} \int_0^t \frac{d}{ds} \|u(s)\|^2 ds + \beta(t+\alpha)\right)^2, \quad t \in [0, T_{\max}).
\end{aligned}
$$

Then it follows from (20) and the above inequality that

$$
\begin{aligned}
(F'(t))^2 &= 4 \left( \frac{1}{2} \int_0^t \frac{d}{ds} \|u(s)\|^2 ds + \beta(t+\alpha) \right)^2 && (23) \\
&\le 4F(t) (\eta^2(t) + \beta), \quad t \in [0, T_{\max}).
\end{aligned}
$$

In view of (21), (22) and (23), we have

$$
\begin{aligned}
F(t)F''(t) - \frac{p}{2}(F'(t))^2 &\ge F(t) \left( 2(\|u_0\|_p^p - \|u_0\|^2) + 2p\eta^2(t) + 2\beta - 2p(\eta^2(t) + \beta) \right) \\
&= F(t) \left( 2(\|u_0\|_p^p - \|u_0\|^2) - 2(p-1)\beta \right), \quad t \in [0, T_{\max}),
\end{aligned}
$$

which is nonnegative if we take $\beta$ small enough such that

$$
0 < \beta \le \frac{\|u_0\|_p^p - \|u_0\|^2}{p-1}. \qquad (24)
$$

Then it follows from Lemma 4 that

$$
\begin{aligned}
T_{\max} &\le \frac{F(0)}{\left(\frac{p}{2}-1\right) F'(0)} \\
&= \frac{1}{p-2} \left( \alpha + \frac{\|u_0\|^2}{\beta\alpha} T_{\max} \right).
\end{aligned}
\qquad (25)
$$

By taking $\alpha$ large enough such that

$$
\alpha > \frac{\|u_0\|^2}{(p-2)\beta}, \qquad (26)
$$

we get from (25) that

$$
T_{\max} \le \frac{\beta \alpha^2}{(p-2)\beta\alpha - \|u_0\|^2}. \qquad (27)
$$

The above analysis shows that $(\rho := \beta\alpha)$

$$
T_{\max} \le \inf_{(\rho, \alpha) \in \Phi} f(\rho, \alpha), \qquad (28)
$$

where

$$
\begin{gather*}
\Phi := \left\{ (\rho, \alpha) : \rho > \frac{\|u_0\|^2}{p-2}, \alpha \ge \frac{(p-1)\rho}{\|u_0\|_p^p - \|u_0\|^2} \right\}, \\
f(\rho, \alpha) := \frac{\rho\alpha}{(p-2)\rho - \|u_0\|^2}.
\end{gather*}
$$
Since $f(\rho, \cdot)$ is increasing, we get

$$
\begin{align*}
T_{\max} &\le \inf_{\rho > \frac{\|u_0\|^2}{p-2}} f\left(\rho, \frac{(p-1)\rho}{\|u_0\|_p^p - \|u_0\|^2}\right) \\
&= \inf_{\rho > \frac{\|u_0\|^2}{p-2}} \frac{(p-1)\rho^2}{((p-2)\rho - \|u_0\|^2)(\|u_0\|_p^p - \|u_0\|^2)} \\
&= \frac{(p-1)\rho^2}{((p-2)\rho - \|u_0\|^2)(\|u_0\|_p^p - \|u_0\|^2)} \Big|_{\rho = \frac{2\|u_0\|^2}{p-2}} \\
&= \frac{4(p-1)\|u_0\|^2}{(p-2)^2(\|u_0\|_p^p - \|u_0\|^2)}.
\end{align*}
$$

So, (18) is true.

(2) **Lower bound estimate.** In the part, we will give the low bound estimate, i.e., show that

$$
T_{\max} \geq \frac{1}{2} \int_{\|u_0\|^2}^{\infty} \frac{d\xi}{\kappa^p \xi^{\frac{p}{2}-2} - \xi}. \qquad (29)
$$

By $(R_2)$, (8), and (12), we have

$$
\begin{align*}
\frac{d}{dt} \|u(t)\|^2 &= -2I(u(t)) \\
&= -2\|u(t)\|^2 + 2\|u(t)\|_p^p \\
&\le -2\|u(t)\|^2 + 2\kappa^p \|u(t)\|^p, \quad t \in [0, T_{\max}).
\end{align*}
$$

Since $I(u(t)) < 0$ (see $(R_3)$), we get $-2\|u(t)\|^2 + 2\kappa^p \|u(t)\|^p \ge -2I(u(t)) > 0$, then the above inequality can be rewritten as

$$
\frac{d}{dt} \|u(t)\|^2 - \frac{2\|u(t)\|^2 + 2\kappa^p \|u(t)\|^p}{\|u(t)\|^2 \|u(t)\|^p} \le 1, \quad t \in [0, T_{\max}). \quad (30)
$$

Since *u* blow up at $T_{\max}$, by Theorem 1,

$$
\lim_{t \uparrow T_{\max}} \|u(t)\|^2 = \infty,
$$

integrating (30) from 0 to $T_{\max}$, we get (29).

<figure><img src="image_4.png" /></figure>

References

[1] G. I. Barenblatt, Y. P. Zheltov, I. N. Kochina, "Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks", *J. Appl. Math. Stochastic Anal.* **24** (1961), p. 1286-1303.

[2] T. B. Benjamin, J. L. Bona, J. J. Mahony, "Model equations for long waves in nonlinear dispersive systems", *Philos. Trans. R. Soc. Lond., Ser. A* **272** (1972), no. 1220, p. 47-78.

[3] M. O. Korpusov, A. G. Sveshnikov, "Three-dimensional nonlinear evolutionary pseudoparabolic equations in mathematical physics", *Zh. Vychisl. Mat. Mat. Fiz.* **43** (2003), no. 12, p. 1835-1869.

[4] S. G. Sajjadi, T. A. Smith, "Model equation for long waves in nonlinear dispersive system on a viscous liquid", *Adv. Appl. Fluid Mech.* **3** (2008), no. 2, p. 219-237.

[5] S. L. Sobolev, "On a new problem of mathematical physics", *Izv. Akad. Nauk SSSR, Ser. Mat.* **18** (1954), p. 3-50.

[6] T. W. Ting, "Certain non-steady flows of second-order fluids", *Arch. Ration. Mech. Anal.* **14** (1963), p. 1-26.

[7] R. E. Showalter, T. W. Ting, "Pseudoparabolic partial differential equations", *SIAM J. Math. Anal.* **1** (1970), p. 1-26.

[8] W. Liu, J. Yu, "A note on blow-up of solution for a class of semilinear pseudo-parabolic equations", *J. Funct. Anal.* **274** (2018), no. 5, p. 1276-1283.

[9] P. Luo, "Blow-up phenomena for a pseudo-parabolic equation", *Math. Methods Appl. Sci.* **38** (2015), no. 12, article no. 10.1002/mma.3253.

[10] G. Xu, J. Zhou, "Lifespan for a semilinear pseudo-parabolic equation", *Math. Methods Appl. Sci.* **41** (2018), no. 2, p. 705-713.
[11] R. Xu, J. Su, "Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations", *J. Funct. Anal.* **264** (2013), no. 12, p. 2732-2763.

[12] R. Xu, J. Su, "Addendum to "Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations"", *J. Funct. Anal.* **270** (2016), no. 10, p. 4039-4041.

[13] R. Xu, X. Wang, Y. Yang, "Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy", *Appl. Math. Lett.* **83** (2018), p. 176-181.

[14] H. Chen, S. Tian, "Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity", *J. Differ. Equations* **258** (2015), no. 12, p. 4424-4442.

[15] X. Wang, R. Xu, "Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation", *Adv. Nonlinear Anal.* **10** (2021), p. 261-288.

[16] W. Lian, J. Wang, R. Xu, "Global existence and blow up of solutions for pseudo-parabolic equation with singular potential", *J. Differ. Equations* **269** (2020), no. 6, p. 4914-4959.

[17] S. Khomrutai, N. Kitisin, "Cauchy problems of sublinear pseudoparabolic equations with unbounded, time-dependent potential", https://arxiv.org/abs/1504.02636v1, 2015.

[18] X. Zhu, F. Li, Y. Li, "Some sharp results about the global existence and blowup of solutions to a class of pseudo-parabolic equations", *Proc. R. Soc. Edinb., Sect. A, Math.* **147** (2017), no. 6, p. 1311-1331.

[19] D. H. Sattinger, "On global solution of nonlinear hyperbolic equations", *Arch. Ration. Mech. Anal.* **30** (1968), p. 148-172.

[20] L. E. Payne, D. H. Sattinger, "Saddle points and instability of nonlinear hyperbolic equations", *Isr. J. Math.* **22**(1975) (1976), article no. 10.1007/BF02761595.

[21] Y. Liu, J. Zhao, "On potential wells and applications to semilinear hyperbolic equations and parabolic equations", *Nonlinear Anal., Theory Methods Appl.* **64** (2006), no. 12, p. 2665-2687.

[22] J. Zhou, "Blow-up and exponential decay of solutions to a class of pseudo-parabolic equation", *Bull. Belg. Math. Soc.* **26** (2019), p. 773-785.

[23] H. A. Levine, "Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_t = Au + F(u)$", *Trans. Am. Math. Soc.* **192** (1974), p. 1-21.