On the global existence and blow-up for the double dispersion equation with exponential term

1.   Introduction

This paper is devoted to the following initial-boundary value problem for the double dispersion equation

where $\Omega$ is a bounded domain of $\mathbb{R}^2$ having smooth boundary $\partial \Omega$ and $u = u(x, t): \Omega\times \mathbb{R}^+ \rightarrow \mathbb{R}$. The damped term $h(u_t)$ is given by $h(u_t) = |u_t|^{q-1}u_t$ with $q > 1$, and the nonlinearity $f\in C^1(\mathbb{R}, \mathbb{R})$ admits

(H1) for each $\beta > 0$, there exists a positive constant $C_\beta > 0$ depending only on $\beta$ such that

On account of the possibility of energy exchange through lateral surfaces of the waveguide in the physical study of nonlinear wave propagation in waveguide, the longitudinal displacement $u(x, t)$ of the rod satisfies the following double dispersion equation (DDE) ^[1,2]

and the general cubic double dispersion equation (CDDE)

Here $a, b, c > 0$ and $d \neq 0$ are some constants depending on the Young modulus, the shearing modulus, the density of the waveguide and the Poisson coefficient.

Due to the wide applications in the real world, the initial value problem and initial-boundary value problem of double dispersion equation have drawn much attention from mathematicians. In ^[3,4,5], the authors studied global solutions with $f'(s)\geq C$ (bounded below) for following the generalized DDE which includes Eq (1.2) as special cases,

Moreover, they also showed the nonexistence of the global solution under some other conditions. When $d = 0$, Liu ^[6] investigated the global existence and nonexistence of solutions for the initial-boundary value problem of (1.3) with $|f(u)|\leq C |u|^{p}$.

For the multidimensional generalized form of (1.3)

Polat ^[7] researched the existence of global solutions also in the cases of $f'(s)\geq C$. Wang ^[8] considered the global existence and asymptotic behavior of the small amplitude solution in the time-weighted Sobolev space for the Cauchy problem of (1.4) with $\left|f(u)\right|\leq C|u|^{\alpha-j}$. Wang ^[9,10] investigated the asymptotic profile of solutions for the Cauchy problem of (1.4) with $f(u) = O(u^2)$. Su ^[11] researched the existence and nonexistence of a global solution in the natural energy space for the initial-boundary value problem of (1.4) with $f(u) = \beta |u|^{p-1}u$. So, motivated by this fact and the so-called Moser-Trudinger type inequalities ^{[12,13,14,15]}, it was natural to consider nonlinearities with exponential growth.

For the generalized double dispersion (1.4) with exponential nonlinearity, Zhang ^[16,17] proved the existence and nonexistence of global weak solution for the Cauchy problem of (1.4) with $d = 0$. Guo ^[18] established the sufficient conditions of finite time blow-up of solutions in the cases of arbitrary positive initial energy for the Cauchy problem of (1.4) with $d = 0$. As far as we are concerned, there are no results on the global existence and finite time blow-up of solutions for the initial-boundary value problem of (1.4) with both nonlinear damped and exponential nonlinearity.

Furthermore, there are many works that focus on the wave equation with exponential nonlinearity. Global well-posedness in the defocusing case was established by Nakamura ^[19] for small data, Atallah ^[20] in the radial case, then by Ibrahim ^[21] and Struwe ^[22,23], see also ^{[24,25,26,27,28,29]} and their references.

It is the aim of this manuscript to obtain results about the global existence and finite time blow-up of solutions under sufficient conditions for the problem (1.1) in the case that $f$ is a source term and admits an exponential growth and $h(u_t)$ is polynomial growth. This is the first attempt in the literature to take into account both nonlinear damped term and the exponential nonlinear source for the problem (1.1).

This paper is organized as follows. We first establish the local well-posedness by the standard Galërkin method and contraction mapping principle(see Theorem 3.1). By means of the potential well theory, we provide the sufficient conditions of global solutions with subcritical initial energy (see Theorem 4.1). In the case of the negative initial energy or the initial data in the potential well, we prove the local solutions will blow up in finite time (see Theorems 5.1 and 5.2). Moreover, we also constructed the sufficient conditions of finite time blow-up of solutions in the case of arbitrary positive initial energy (see Theorem 5.3).

It is worth mentioning here that the sufficient conditions of blow-up of solutions can be investigated based on the concave method ^[17,18,27] in the absence of the nonlinear damped term $h(u_t)$ or $h(u_t) = u_t$ in (1.1). However, when nonlinear damping and exponential source terms are both present in the equation, it seems that their method does not work on our problem directly. We research the blow-up of solutions in the cases of $E(0) < 0$ or subcritical initial energy mainly by exploiting an argument from the one devised in ^[30,31,32], which investigated the blow-up of solutions to the wave equations with polynomial-type nonlinearity of form $|s|^ms$, and the case of $E(0) > 0$ mainly by ^[33].

We conclude this section with several notations given. The notation $(\cdot, \cdot)$ stands for the $L^2$-inner product and $\langle\cdot, \cdot \rangle$ is used for the notation of duality pairing between dual space. For brevity, we use the same letter $C$ to denote different positive constants, and $C(\cdot \cdot \cdot)$ to denote positive constants depending on the quantities appearing in the parenthesis.

2.   Preliminaries

For brevity, we use the following abbreviations:

with $1\leq p\leq \infty$. $H^{-1} = H^{-1}(\Omega)$ is the dual space of $H_0^1$. Let $A = -\Delta$. Then, $\langle Au, v\rangle = (\nabla u, \nabla v)$, for $u, v\in H_0^1$ and the domain of $A$ is $D(A) = H^2\cap H_0^1;$ $A$ is a positive, self-adjoint and invertible operator and the inverse operator $A^{-1}$ is compact ^[34]. Consequently, the operator $A$ possesses an infinitely countable positive eigenvalues:

and a corresponding sequence of eigenfunctions $\{e_j: j = 1, 2, \cdots\}$ that forms an orthogonal basis for $L^2$. Also, the sequence $\{e_j: j = 1, 2, \cdots\}$ is an orthogonal basis for $H_0^1$. In addition, the linear span of $\{e_j: j = 1, 2, \cdots\}$ is dense in $L^p$ for any $1 \leq p < \infty.$ Since the domain $\Omega$ is smooth, then $e_j\in C^\infty(\Omega)$.

For any $u \in L^2$, there exists $u_j = u_j(t) = (u, e_j)$ such that

The powers of $A$ are defined as follows ^[34]

For $s\geq 0$, $A^s: D(A^s)\subset L^2 \rightarrow L^2,$ and the domain of $A^s$ is given by

which endowed with the graph norm

and the associated scalar product

where $v_j = (v, e_j)$. Especially, $D(A^s) = H_0^{2s}$ for $\frac{1}{4} < s\leq \frac{1}{2}$ (see ^[35]).

It worth to be mentioned that we introduce the space $H_s$ (with $s > 0$) as the domain $D(A^s)$ equipped with the graph norm $\|u\|_{D(A^s)} = \|A^s u\|$. If $s$ is negative, we define $H_s$ as the completion of $L^2$ with respect to the norm $\|u\|_{D(A^s)} = \|A^{-|s|} u\|$.

By the Poincaré inequality, one can easily see that if $u\in L^2$ then $(-\Delta)^{-\frac{1}{2}}u\in L^2$. So, we can define a Hilbert space as follows

where the scalar product $(\cdot, \cdot)_{\mathcal{H}}$ is defined by

and the norm $\|\cdot\|_{\mathcal{H}}$ is defined by

We note that the Poincaré inequality implies that the norms $\|\cdot\|$ and $\|\cdot\|_{\mathcal{H}}$ are equivalent norms on $L^2$.

Applying the operator $(-\Delta)^{-1}$ to (1.1). Then (1.1) becomes

Since $(-\Delta)^{-1}\Delta^2 u = -\Delta u$, for any $u\in \left\{u\in H^4\cap H_0^1: \Delta u\mid_{\partial\Omega} = 0\right\}$ (see [34,Lemma 1.7]), then the Eq (2.2) can be written as

Then, the weak solution of (2.2) with the initial data of (1.1) and boundary value condition $u\mid_{\partial\Omega} = 0$ is said to be the weak solution of the problem (1.1). This leads to the following definition.

Definition 2.1 (weak solution). A function $u\in C\left([0, T]; H_0^1\right)\cap C^1\left([0, T]; L^2\right)$ with $u_t\in L^{q+1}((0, T)\times \Omega)$ is said to be a weak solution to the problem (1.1) over $[0, T]$, if and only if for any $t\in [0, T]$, it satisfies

for all test functions $\varphi\in H_0^1$, and

Now, we give the Trudinger-Moser inequality ^[14,15] which will be used repeatedly to estimate the exponential nonlinearity.

Lemma 2.1. For all $u\in W_0^{1, n} (n\geq 2)$

and there exist positive constants $C(n)$ which depends on $n$ only, such that

where $|\Omega| = \int_{\Omega} \mathrm{d} x$, $\alpha(n) = nw_{n-1}^{\frac{1}{n-1}}$, and $w_{n-1}$ is the $(n-1)$-dimensional measure of the $(n-1)$-sphere.

Remark 2.1. From Lemma 2.1, we know that $f(u)\in L^1$ (in fact, from (2.4), $f(u)\in L^p$ for any $p\geq 1$) for all $u\in H_0^{1}$, and let

we also infer that $F(u)\in L^1$ and

The following lemma is for the convergence of the approximate solutions.

Lemma 2.2 (^[36]). Let $X$ and $Y$ be Banach spaces, $X'$ and $Y'$ be the dual spaces of $X$ and $Y$ respectively, and $X$ a dense subset of $Y$ and the inclusion map of $X$ into $Y$ continuous. Assume that

then $\chi = u_t$ in $L^\infty(0, T; Y')$.

The next lemma is straightforward and follows from the continuity and monotonicity of the function $h(u) = |u|^{q-1}u = |u|^{q} sgn(u)$.

Lemma 2.3. Let $h(u) = |u|^{q-1}u = |u|^{q} sgn(u)$. Then $h$ generates a monotone operator from $L^{1+\frac{1}{q}}(\Omega)$ into $L^{1+\frac{1}{q}}(\Omega)$, i.e.,

Moreover, the mapping $\lambda\mapsto \langle h(u+\lambda v), \eta \rangle$ is continuous from $\mathbb{R}$ to $\mathbb{R}$ for every fixed $u, v, \eta \in L^{q+1}$.

The continuity of the solutions needs the following result which can be found in ^[37].

Lemma 2.4. Let $V$ and $Y$ be Banach spaces, $V$ reflexive, $V$ a dense subset of $Y$ and the inclusion map of $V$ into $Y$ continuous. Then,

where

3.   Local solutions

This section focus on the local well-posedness of (1.1) in the natural energy space. First, we establish the existence of a local weak solution to the corresponding linear problem of (1.1) by using a standard Galerkin approximation scheme based on the eigenfunctions $\{e_j\}^{\infty}_{j = 1}$ of the operator $A = -\Delta$. The well-posedness of the nonlinear problem (1.1) shall be researched by the contraction mapping principle, and mainly by the Trudinger-Moser inequality to deal with the nonlinearity.

Theorem 3.1. Let $f$ satisfy (H1), $u_0\in H_0^1$, and $u_1\in L^2$. Then there exists a unique weak solution $u$ of (1.1) in $\Omega\times (0, T_{\max})$, where $T_{\max}$ is the life time of solutions. In addition, $u$ satisfies the energy identity

where

Here and after $\|\cdot\|_{\mathcal{H}}$ is denoted by (2.1). Moreover, if

then $T_{\max}=\infty$.

In order to prove Theorem 3.1, we firstly consider the following auxiliary result.

Lemma 3.1. Let $T > 0$, $u_0\in H_0^1$, $u_1\in L^2$, and $u\in C\left([0, T]; H_0^1\right)$ with the norm $\max\limits_{t\in[0, T]}\|u\|_{H^1_0}\leq R$ for some constant $R > 0$. Assume that $f$ satisfies (H1). Then, there exists a unique

which solves the linear problem

Proof. Let $\{e_j\}^{\infty}_{j = 1}$ be the orthonormal basis for $L^2$, as described in Section 2. For $m\in \mathbb{N}$ given, we seek $m$ functions $g_{1, m}, g_{2, m}, ..., g_{m, m}\in C^2[0, T]$ such that

solves the problem

for every $\eta \in \text{Span}\{e_1, e_2, \ldots, e_m\}$. Taking $\eta = e_j$ in (3.4) yields the following Cauchy problem for a linear ordinary differential equation with unknown $g_{j, m}$:

Then, by the standard ordinary differential equations theory, for all $j$, the above Cauchy problem yields a unique global solution $g_{j, m}\in C^2(0, t_m)$. In turn, this gives a unique $v_m(t)\in H^2\cap H_0^1$ for every $t\in(0, t_m)$ defined by (3.3) and satisfying (3.4)–(3.6). Taking $\eta = v_{mt}$ in (3.4), and integrating over $[0, t] \subset (0, t_m)$, we obtain

We estimate the last term in the right-hand side of (3.7) thanks to the assumption (H1), the Hölder inequality. More precisely,

By means of the Young inequality, we achieve

for a appropriate small constant $0 < \varepsilon < 1$. Since the assumption on $u$ that $\max\limits_{t\in[0, T]}\|u\|_{H^1_0}\leq R$, we find

With the Moser-Trudinger inequality, for each $\beta < \frac{4q\pi}{(q+1)R^2}$, we get

Thus,

From the facts (3.5)–(3.8) we obtain

for every $m\geq 1$, where $C_T > 0$ is independent of $m$. So we can extend the approximated solutions $v_m$ to the whole interval $[0, T]$ and the uniform estimate (3.9) also holds on $[0, T]$. Besides we get

By the definition of $h(v_{mt})$, we have

That means

Moreover, by the Eq (3.4) we know that

Therefore, we can extract from the sequence $\{v_m\}$ a subsequence which we still denote by $\{v_m\}$ such that

Since $n = 2$ and $\Omega$ is bounded we deduce immediately that $u\in H_0^1\hookrightarrow L^r$ for all $1\leq r < +\infty$. From Lemma 2.2, we have that

Integrating (3.4) with respect to $t$ from $0$ to $t$ and let $m\rightarrow \infty$ we obtain

for all test functions $\eta\in H_0^1$. So, we get a solution $v$ to the following initial-boundary problem

and

Consequently,

By making use of Lemma 2.4, we deduce immediately that

Noting that

then form Corollary 4.1 of Starauss in ^[37], we conclude that $\|v_t\|^2+\|\nabla v\|^2$ is a continuous function of $t$. Combining with (3.12), we have

So far, we shall have proved the existence of the solution of (3.2), if we show that $\chi = h(v_t)$. Observing the monotonicity and continuity of $h(v_t)$, we follow the idea of monotonicity argument used in ^[38].

On the one hand, noting that

and by considering ideas from [37,Theorem 4.1], $v$ satisfies for all $t\in(0, T]$ the energy identity

On the other hand, it follows from (3.7), (3.10) and (3.11) that

Therefore, (3.13) and (3.14) yield

Now, let $\varphi \in L^{1+\frac{1}{q}}(0, T\times \Omega)$ be arbitrary. Then it follows from Lemma 2.3 and (3.15) that

By choosing $\varphi(t) = v_t-\lambda \psi(t)$, where $\lambda\in \mathbb{R}$, (3.16) yields

for all $\lambda\geq 0$ and $\psi \in L^{q+1}(0, T\times \Omega)$. By letting $\lambda\rightarrow 0$ and recalling the continuity of $h$, we achieve

which implies $\chi = h(v_t)$.

The last work is the uniqueness of solutions, which follows arguing for contradiction: if $v$ and $w$ are two solutions of (3.2) which share the same initial data, by subtracting the equations and $v$ is substituted by $v-w$, we would get

Observing the Mazur inequality

we get

So, (2.1) can be estimated as

which immediately yields $w\equiv v$. The proof of Lemma 3.1 is now completed.

Now we turn to the nonlinear problem (1.1) by using the contraction mapping principle.

Proof of Theorem 3.1 Taking $(u_0, u_1)\in H_0^1\times L^2$, let $R^2 = 2(\|u_0\|_{H_0^1}^2+\|u_1\|^2_{\mathcal{H}})$ and for any $T > 0$ considering a closed convex set $\mathcal{M}_T$ to be a ball of radius $R$ in the space $\mathcal{M} = C\left([0, T]; H_0^{1} \right)\cap C^1\left([0, T]; L^2 \right)$,

with the norm $\|u\|^2_{\mathcal{M}_T} = \max\limits_{t\in[0, T]}\left\{\|u\|^2_{H_0^1}+\|u_t\|^2_{\mathcal{H}}\right\}$ and metric $d(u, v) = \|u-v\|_{\mathcal{M}_T}$. Obviously, $(\mathcal{M}_T, d)$ is a complete metric space. By Lemma 3.1, for any $u\in \mathcal{M}_T$ we may define $v = \Phi(u)$, being $v$ the unique solution to problem (3.2). We claim that, for a suitable $T > 0$, $\Phi$ is a contractive map satisfying $\Phi(\mathcal{M}_T) \subseteq \mathcal{M}_T$.

It follows from the fact (3.13) that

For the last term, we argue in the same spirit (although slightly differently) as for (3.8) and we get

Combining (3.19) with (3.20) and taking the maximum over $[0, T]$ gives

Choosing $T$ sufficiently small, we get $\left\|v\right\|_{\mathcal{M}_T} \leq R$, which shows that $\Phi(\mathcal{M}_T)\subseteq \mathcal{M}_T$. Now, taking $w_1$ and $w_2$ in $\mathcal{M}_T$, subtracting the two equations (3.2) for $v_1 = \Phi(u_1)$ and $v_2 = \Phi(u_2)$, and setting $z = v_1-v_2$ then $z$ is the unique solution of the problem

Taking the assumption (H1) into account, we have

Using (3.18) again it holds

By (3.21), Lemma 2.1, the Hölder inequality and the Sobolev imbedding inequality, it deduces that

for each $\beta < \frac{2(q-1)\pi}{(q+1)R^2}$, where we have used the fact that $\frac{1}{2}+\frac{1}{q+1}+\frac{q-1}{2(q+1)} = 1$. Using the Young inequality, the estimate (3.22) becomes

Consequently,

Applying Gronwall inequality and taking the maximum over $[0, T]$, it follows that

for some $\delta < 1$ provided $T$ is sufficiently small. By the contraction mapping principle, there exists a unique solution $u$ to (1.1) defined on $[0, T]$, $u\in C\left([0, T]; H_0^{1} \right)\cap C^1\left([0, T]; L^2 \right)$ and from (3.19) the energy equality (3.1) holds. Theorem 3.1 is proved.

4.   Global solutions

The goal of this section is to prove that the local solution established in Theorem 3.1 can be extended globally in time when the initial data inside the potential well. So that we firstly introduce Nehari functional, Nehari manifold and the stable sets. Let

and

Related to the functional $J$, we have the well known Nehari manifold

Let $f :\mathbb{R}\rightarrow \mathbb{R}$ is a $C^1$ function verifying both the assumption (H1) and the following properties:

(H2) The function $f$ satisfies the following condition near the origin

(H3) There exists $\theta > 2$ such that

(H4) There are constants $R_0, M_0 > 0$ such that, for all $|t|\geq R_0$,

Then, it can be checked that the mountain-pass level $d$ can be characterized as

The proof is refered to ^[17,27].

Hereafter, we will denote by $W$ and $V$ the following sets:

Under the assumptions imposed on $f$, it is can be proved that $W$ and $V$ are invariant sets related to (1.1) (Lemmas 4.1 and 4.1), which provide the possibilities for us to establish the global existence and nonexistence of the solutions.

Remark 4.1. An interesting question is whether there is a function that meets the conditions (H1)–(H4). Now, we construct explicitly an example such that all assumptions (H1)–(H4) are satisfied.

Example 1. Let $f(t) = |t|^{p-1}te^{\frac{\beta}{2} t^2}$ with $p > 1$. Then $f(t)$ satisfies (H1)–(H4).

Proof. It follows from the fundamental inequality

that

Which shows $f(t)$ satisfies (H1). Obviously, $f(t)$ satisfies (H2).

For (H3), by the definition of $F(t)$, we get

For $p > 1$, we have

Thus, $f(t)$ satisfies (H3).

At last, we show $f$ verifies (H4). It follows from (4.2) that

Obviously, for any constant $R_0 > 0$, $|t|^{-1} < \frac{1}{R_0}$ for $|t| > R_0$, and $\frac{2n}{\beta(2(n-1)+p+1)} < \frac{1}{\beta}$. Thus,

where $M_0 = \frac{1}{\beta R_0}$. $f$ satisfies (H4).

The following result is about the invariance of the set $W$ under the flow of (1.1).

Lemma 4.1 (^[17,27]). Let $f$ satisfy (H1)–(H4) and $u$ be the unique local solution to (1.1). Assume that $E(0) < d$, then for all $t\in[0, T]$, $u$ belongs to $W$ provide that $u_0$ belongs to $W$, and

Now we concerned with the global existence of the solution for the problem (1.1).

Theorem 4.1 (Global solutions for $E(0) < d$). Let $f(u)$ satisfy (H1)–(H4), and $u$ be the unique local solution to (1.1). Assume that $E(0) < d$, and $I(u_0) > 0$ or $\|u_0\|_{H_0^1} = 0$. Then, $u$ exists globally and $u\in W$, for all $t\in[0, \infty)$.

Proof. From (4.1) we have

Since $E(0) < d$ and $I(u_0) > 0$ or $u_0 = 0$, it follows from Lemma 4.1 that $I(u) > 0$ or $\|u\|_{H_0^1} = 0$ for all $t\in [0, T_{\max})$, which implies that

for all $t\in[0, T_{\max})$ and $\theta > 2$. Therefore, by virtue of Theorem 3.1 it yields $T_{\max} = \infty$.

5.   Blow-up of solutions

This section is concerned with results on finite time blow-up of the solutions for the problem (1.1) with $f(u) = |u|^{p-1}ue^{{\frac{\beta}{2}}|u|^2}, p > 1$. We firstly prove that when the initial energy is negative, the local solution can not be extended globally in time. Secondly, in the case of $E(0) < d$, we prove that when the initial data $u_0\in V$, the local solution blows up in finite time. Lastly, we construct the sufficient conditions of finite time blow-up of the solutions with arbitrary positive initial energy.

Now, we state the main results on the finite time blow-up of the solutions.

Theorem 5.1 (Blow-up for $E(0)\leq0$). Let $f(u) = |u|^{p-1}ue^{{\frac{\beta}{2}} |u|^2}$ for $p > 1$, and $u$ be the unique local solution to (1.1). Assume that $E(0) < 0$ or $E(0) = 0$ and $(u_0, u_1)_{\mathcal{H}} > 0$. Then, the problem (1.1) does not admit any global weak solution.

From Theorem 5.1 we can easily obtain the following result.

Corollary 5.1. Let $f(u) = |u|^{p-1}ue^{{\frac{\beta}{2}} |u|^2}$ with $p > 1$ and $u$ be the unique local solution to (1.1). If there exists $t_0\in[0, T]$ such that $E(t_0) < 0$, then the problem (1.1) does not admit any global weak solution.

The result about finite time blow-up of the solutions with the subcritical initial energy is based on the following three lemmas.

Lemma 5.1 (^[17,27]). Let $f(u) = |u|^{p-1}ue^{{\frac{\beta}{2}} |u|^2}$ with $p > 1$ and $u$ be the unique local solution to (1.1). Assume that $E(0) < d$, then for all $t\in[0, T]$, $u$ belongs to $V$ provide that $u_0$ belongs to $V$, and

Lemma 5.2. Let $f(u) = |u|^{p-1}ue^{{\frac{\beta}{2}} |u|^2}$ with $p > 1$ and

Then $\varphi(\lambda)$ is strictly increasing on $0 < \lambda < \infty$.

Proof. By a simple calculation, we check

and the equality holds only for $u = 0$. With (5.2) we get

the equality holds only for $u = 0$. Thus, the proof of Lemma 5.2 is completed.

Lemma 5.3. Let $f(u) = |u|^{p-1}ue^{{\frac{\beta}{2}} |u|^2}$ with $p > 1$ and $u$ be the unique local solution to (1.1). Assume that $u$ belongs to $V$ then

Proof. We define the function $W(\lambda) = I(\lambda u)$ for $\lambda > 0$. Observe that $W(1) = I(u) < 0$ and

for $\lambda$ sufficiently small. Hence there exists some $\lambda_0\in(0, 1)$ such that $W(\lambda_0) = I(\lambda_0u) = 0$. That is $\lambda_0^2\|u\|^2_{H_0^1} = \int_{\Omega} \lambda_0 uf(\lambda_0 u)\mathrm{d} x$. By the definition of $d$, we obtain

It follows from Lemma 5.2 that

That is,

This completes the proof of Lemma 5.3.

Theorem 5.2 (Blow-up for $E(0) < d$). Let $f(u) = |u|^{p-1}ue^{{\frac{\beta}{2}} |u|^2}$ with $p > 1$ and $u$ be the unique local solution to (1.1). Assume that $E(0) < \gamma d$ for some constant $0 < \gamma\leq\frac{1}{2}$, and $u_0\in V$. Then, the problem (1.1) does not admit any global weak solution.

Theorem 5.3 (Blow-up for $E(0) > 0$). Let $f(u) = |u|^{p-1}ue^{{\frac{\beta}{2}} |u|^2}$ with $p > 1$ and $u$ be the unique local solution to (1.1). Assume that $E(0) > 0$, and

then the problem (1.1) does not admit any global weak solution, where $M$ is the root of the equation

on $\left(M_0, +\infty\right)$,

$C_{\beta, \Omega} > 0$ is a constant depending on $\beta$ and $\Omega$, and $\lambda_1$ is the first eigenvalue of $-\Delta$ with zero Dirichlet boundary data on the smooth bounded domain $\Omega\subset \mathbb{R}^2$.

Now, we concerned with our proofs of Theorems 5.1–5.3.

Proof of Theorem 5.1 Arguing by contradiction, we suppose that $T_{\max} = +\infty$. For any $0 < T < \infty$, we define

Then,

Note that the standard approximation argument shows that $G''(t)$ exists and by the Eq (2.3), we have

Let

It follows from the energy equality (3.1) that

So that $H(t)$ is increasing on $[0, T]$ and noting the fact (4.3), we have

Now considering the following function defined on $[0, T]$

where $0 < \alpha < 1$ and $\epsilon > 0$ are sufficiently small and given later. A simple computation entails

By the Young inequality we have

for some $\delta > 0$ to be fixed later. Substituting (5.7) into (5.6), and noting $H(t) = -E(t)$ we obtain

Taking $\delta$ such that $\delta^{-\frac{q+1}{q}} = kH^{-\alpha}(t)$, the constant $k > 0$ will be chosen later, then from (5.5), we have

For any $m > q$, by means of the Hölder inequality, we obtain

Taking $m > p$, it holds

Then, taking $m > \max\{p, q\}$, we arrive at

Thus,

Taking $0 < \alpha\leq \frac{m-q}{q(m+1)}$ such that $\alpha q+\frac{q+1}{m+1}\leq 1$ and $s = \alpha q+\frac{q+1}{m+1}-1\leq0$. By (5.5) again,

Therefore, (5.8) can be estimated by

Taking $k > 0$ sufficiently large such that $1-\frac{2}{p+1}-C_{\Omega, \beta}(p+1)^{\frac{q+1}{m+1}- 1} k^{-q} H(0)^s > 0$, and for the fixed $k$ picking $\epsilon > 0$ appropriate small such that $(1-\alpha)-\frac{2\epsilon kq}{q+1} > 0$ and

for $H(0) > 0$. Thus, we have

On the other hand, by the Hölder inequality and the Young inequality, we have

for $0 < \alpha < \frac{1}{2}$.

Case 1. When $\|u\|\geq1$, taking $\alpha\leq\frac{p-1}{2(p+1)}$ such that $\frac{2}{1-2\alpha}\leq p+1$, then by the Hölder inequality, we achieve

Case 2. When $\|u\|\leq1$, it follows from the energy equality (3.1), the assumption $E(0) < 0$ and the last inequality of (5.5) that

Noting that $\frac{2}{1-2\alpha} > 2$, then

Therefore, we obtain

Consequently, by taking $0 < \alpha\leq\min\left\{\frac{m-q}{q(m+1)}, \frac{p-1}{2(p+1)}\right\}$ and from the above argument we have

Combining with (5.11), it holds that

The fact (5.10) and the assumption $(u_0, u_1)_{\mathcal{H}} > 0$ if $E(0) = 0$ entail $L(0) > 0$ for $E(0)\leq 0$. Then by a calculation we have

which shows that $L(t)$ blows up in finite time

This completes the proof of Theorem 5.1.

The proof of Theorem 5.2 Arguing by contradiction, we suppose that $T_{\max} = +\infty$. Taking $d_1\in(E(0), \gamma d)$ for $0 < \gamma\leq \frac{1}{2}$ and setting

It follows from the energy equality (3.1) that

So that $\tilde{H}(t)$ is increasing on $[0, T]$. Noting that $u(t)\in V$ we conclude from (5.1) that

which deduce that

Considering the following function defined on $[0, T]$

where $G(t)$ is defined by (5.4), $0 < \alpha < 1$ and $\epsilon > 0$ are sufficiently small and given later. By the direct calculation as (5.6), we deduce that

It follows from Lemma 5.3 that

for $0 < \gamma\leq \frac{1}{2}$. Substituting the above inequality into (5.12), it holds

The remainder of the argument is analogous to that in (5.6) and so omitted. The proof Theorem 5.2 is completed.

The proof of Theorem 5.3 Arguing by contradiction, we suppose that $T_{\max} = +\infty$. For $t\in[0, +\infty)$, considering the function

where $G(t)$ is given by (5.4). Similar to the proof of Theorem 5.1, by the estimate (5.7) with $\delta^{-\frac{q+1}{q}} = M^{\frac{1}{q}}$, we have

Case 1. For $q\geq p$, by means of the fact (5.9), and recalling the assumption (4.3), we have

Taking $M > M_0 = \frac{(p+1)C_{\beta, \Omega}}{(q+1)(p-1)}$ such that $(p+1)\left(1-\frac{C_{\beta, \Omega}}{(q+1)M}\right)-2 > 0$, and by using the Poincaré inequality, we deduce

By using the Cauchy inequality,

we get

By a simple calculation, we have

Obviously, there exists $M > M_0$ such that

and the estimate (5.13) becomes

The condition (5.3) guarantees $L_1(0) > 0$. Thus, it yields

By the assumption that $u$ is the global solution, we have, from Corollary 5.1, we have $0\leq E(t)\leq E(0)$. Thus,

Therefore,

Case 2. For $q < p$, observe that the function

is convex. By the properties of convex functions, we have

Thus, by the the Poincaré inequality we have

where $K(M) = p+1-\frac{q-1}{M(p-1)} > 0$ and

By a similar argument to that in Case1, we can obtain (5.14), and in order to avoid redundancy, we omit it here.

On the other hand, it follows from the Hölder inequality and the Poincaré inequality that

which is a contradiction with (5.14). The proof of Theorem 5.3 is completed.

Remark 5.1. Asymptotic behavior of solutions for the problem (1.1) is also an interesting and important work, which is the further work to be considered.

Acknowledgments

The authors thank the referees for their valuable comments and suggestions which helped improving the original manuscript.

The project is supported by the Natural Science Foundation of Henan (202300410109), the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology (2018QNJH19), the training plan for young backbone teachers of Henan University of Technology, the Innovative Funds Plan of Henan University of Technology (2020ZKCJ09).

Conflict of interest

The authors declare there is no conflicts of interest.