Existence and nonexistence of global solutions for logarithmic hyperbolic equation

1.   Introduction

In this paper, we study the initial-boundary value problem for logarithmic hyperbolic equation of $p-$Laplacian type

where

$\Omega\subset R^N$ is a bounded domain with smooth boundary $\partial\Omega$, and parameter $p$ fulfills

The logarithmic nonlinearity arises in a lot of different areas of physics such as quantum mechanics and inflation cosmology, and is applied to nuclear physics, optics and geophysics ^{[1,2,3,4,5,6,7,8]}. Because of these special physical meanings, the research of evolution equations with logarithmic nonlinearity has attracted much attention.

When the logarithmic term in Eq (1.1) is replaced by nonlinear source term $|u|^{r-2}u$, the Eq (1.1) becomes

In the case of $p = 2$ and $r > 2$, J. Ball ^[9] and M. Tsustsumi ^[10] obtained the blow-up solutions with negative initial energy. By using the concavity method, H. A. Levine and L. E. Payne ^[11] established the nonexistence of global weak solutions for Eq (1.6) with the conditions (1.2) and (1.3). Y. J. Ye ^[12] proved the global existence and blow-up result of solutions to the Eq (1.6) with initial-boundary value conditions. S. Ibrahim and A. Lyaghfouri ^[13] considered the Cauchy problem of (1.6), under appropriate assumptions, they established the finite time blow-up of solutions and, hence, extended a result by V. A. Galaktionov and S. I. Pohozaev ^[14]. For the Eq (1.6) with dissipative term, Y. J. Ye ^[15,16] studied the global solutions by constructing a stable set in $W^{1, p}_0(\Omega)$ and the decay property of solution by using an integral inequality ^[17].

S. A. Messaoudi and B. S. Said-Houari ^[18] considered the nonlinear hyperbolic type equation

where $a, b > 0, \alpha, \beta, m, p > 2$ and $\Omega$ is a bounded domain in $R^N (N\geq1)$. Under suitable conditions on $\alpha, \beta, m, p$, they proved a global nonexistence result of solutions with negative initial energy. For the nonlinear wave equation of $p$-Laplacian type

C. Chen et al.^[19] obtained the global existence and uniqueness of solutions and established the long-time behavior of solutions.

L. C. Nhan and T. X. Le ^[20] studied the existence and nonexistence of global weak solutions for a class of $p$-Laplacian evolution equations with logarithmic nonlinearity and gave sufficient conditions for the large time decay and blow-up of solutions. Later, Y. Z. Han et al. ^[21] also considered this problem. They studied global solutions and blow-up solution for arbitrarily high initial energy. For a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic term, under various assumptions about initial values, H. Ding and J. Zhou ^[22] proved the solution exists globally and blow up in finite time. Moreover, T. Boudjeriou ^[23] was concerned with the fractional $p$-Laplacian with logarithmic nonlinearity, by applying the potential well theory and a differential inequality, he proved the existence and decay estimates of global solutions and obtained the blow-up result of solutions.

T. Cazenave and A. Haraux ^[8] considered the following logarithmic wave equation

they gave the existence of solutions for the Cauchy problem of Eq (1.7). P. Gorka ^[4] obtained the global existence of weak solutions for the initial-boundary value problem of Eq (1.7). K. Bartkowski and P. Gorka ^[7] proved the existence of classical solutions and weak solutions for the corresponding one dimensional Cauchy problem of Eq (1.7). In the case of $0 < {\mathcal E}(0)\leq d$, W. Lian et al. ^[24] proved the global existence of solution and obtained the blow-up of solution for the Eq (1.7) with the conditions (1.2) and (1.3).

In this paper, by means of the potential well theory and the concavity analysis method ^{[25,26,27,28,29]}, we prove the global existence and blow-up of solutions of the problem (1.1)–(1.3).

For simplicity, we denote $L^p(\Omega)$ and $L^2(\Omega)$ norm by $\|\cdot\|_p$ and by $\|\cdot\|$ respectively. The space $W^{1, p}_0(\Omega)$ norm $\|\cdot\|_{W^{1, p}_0(\Omega)}$ is replaced by $\|\nabla\cdot\|_p$.

2.   Preliminaries

At first, we define the weak solutions of the problem (1.1)–(1.3) and give a few known lemmas.

Definition 2.1. ^[30] If

and satisfies

then $u(t)$ is called a weak solution of (1.1)–(1.3) on $[0, T)$, where $\varphi\in W_0^{1, p}(\Omega)$.

Lemma 2.1. ^[31] Let $q$ be a real number with $2\leq q < +\infty$ if $2\leq n\leq p$ and $2\leq q\leq\frac{np}{n-p}$ if $2 < p < n$. Then there exists a positive constant $C$ depending on $\Omega, p$ and $q$ such that $\|u\|_q\leq C\|\nabla u\|_p$.

Lemma 2.2. ^[30] Let $B_0, B, B_1$ be Banach spaces with $B_0\subseteq B\subseteq B_1$ and

Suppose that $B_0$ is compactly embedded in $B$ and that $B$ is continuously embedded in $B_1$, then (i) the embedding of $X$ into $L^p(0, T; B)$ is compact if $p < +\infty$. (ii) the embedding of $X$ into $C([0, T]; B)$ is compact if $p = +\infty$ and $q > 1$.

Lemma 2.3. ^[32] Assume that $u_n(x)$ is a bounded sequence in $L^q(\Omega), 1\leq q < +\infty$, $u_n(x)\rightarrow u(x)$ a.e.. Then $u(x)\in L^q(\Omega)$ and $u_n(x)\rightarrow u(x)$ weakly converges in $L^q(\Omega)$.

Lemma 2.4. ^[33,34,35] ($L^2-$logarithmic Sobolev inequality) If $v\in H_0^1(\Omega)$, then

In order to deal with the logarithmic term $|u|^{p-2}u\ln|u|$ in Eq (1.1), we introduce the following $L^p-$logarithmic Sobolev inequality.

Lemma 2.5. ^[36] Let $u\in W_0^{1, p}(\Omega)$, then one has the inequality

where $a > 0$ is a constant.

For convenience, in the following we are going to give the proof of Lemma 2.5.

Proof. By (2.1) in Lemma 2.4, we have

Let $v = u^{\frac p2}$ in (2.3), then we obtain

By direct computation, we get

From (2.5) and Hölder inequality, we receive

By Young inequality $XY\leq \frac{X^\alpha}{\alpha}+\frac{Y^\beta}{\beta}$ with $\alpha = \frac p{p-2}, \ \beta = \frac p2$, we conclude that

It follows from (2.4), (2.6) and (2.7) that

which implies

This completes the proof of Lemma 2.5.

Next, we define the following functionals

and

for $u\in W_0^{1, p}(\Omega)$. By (2.8) and (2.9), we have

We denote the energy functional by

for $u\in W_0^{1, p}(\Omega), \ t\geq0$.

is the initial total energy.

Moreover, we define the Nehari manifold ^[37]

the stable set

and the unstable set

where

It is readily seen that the potential well depth $d$ defined in (2.13) can also be characterized as

Lemma 2.6. Let $u\in W_0^{1, p}(\Omega)$ and $\|u\|_p\not = 0$, then we have

Proof. (a) For $u\in W_0^{1, p}(\Omega)$,

It is easy to get from $\|u\|_p\not = 0$ that (a) is valid.

(b) An elementary calculation shows that

Let $\frac{d}{d\theta}{\mathcal J}(\theta u) = 0$, then we have

It follows from (2.9) that

From (2.16), (2.17) and (2.18), the Eq (2.15) holds.

Lemma 2.7. Suppose that $u\in W_0^{1, p}(\Omega)$ and $\|\nabla u\|_p\not = 0$. Then $d\geq M$, where $M = \frac1{p^2}(2\pi)^{\frac n2}e^{\frac{2(n+p)-p^2}2}$.

Proof. From Lemma 2.6 and Eq (2.10), one has

We get from Lemma 2.5 that

Choosing $a = \sqrt{2\pi}$, we have

It follows from ${\mathcal K}(\theta_*u) = 0$ and (2.20) that

which implies

Thus, we obtain from (2.19) and (2.21) that

Thus, by (2.13) and (2.22), we conclude that $d\geq M > 0$.

3.   Existence of global solutions

In this section, we state and prove the global existence result for the problem (1.1)–(1.3).

Theorem 3.1. Assume that $p$ satisfies (1.5). If $u_0\in W_0^{1, p}(\Omega), \ u_1\in L^2(\Omega)$ and $0 < {\mathcal{E}}(0) < M, \ {\mathcal{K}}(u_0)\geq0$, then there is a global weak solution $u(x, t)$ to the problem (1.1)–(1.3) which meets $u(x, t)\in L^\infty([0, +\infty); W_0^{1, p}(\Omega)), \ u_t(x, t)\in L^\infty([0, +\infty); L^2(\Omega))$.

Proof. Assume that $\{\omega_j\}_{j = 1}^{\infty}$ is a basis of space $W_0^{1, p}(\Omega)$ and that $V_k$ is the subspace of $W_0^{1, p}(\Omega)$ generated by $\{\omega_1, \omega_2, \cdots, \omega_m\}, \ m = 1, 2, \cdots$. We shall look for the approximate solutions $u_m(t) = \sum\limits_{j = 1}^{m}g_{jm}(t)\omega_j$ with $g_{jm}(t)\in C^2[0, T], \ \forall T > 0$. Here the functions $g_{jm}(t)$ fulfil the following system of equations

with initial data

Because $W_0^{1, p}(\Omega)$ is dense in $L^2(\Omega)$, so there exist $\alpha_{jm}$ and $\beta_{jm}$ such that

By Picard's iteration method, the solutions $g_{jm}(t)$ for the Cauchy problem (3.1)–(3.2) exist in $t\in[0, t_m), \ t_m \leq T.$ By the uniformly boundedness of functions $g_{jm}(t)$ and the extension theorem, these solutions $g_{jm}(t)$ exists in the whole interval $[0, T]$.

Multiplying both sides of (3.1) by $g_{jm}'(t)$, summing on $j$ from $1$ to $m$ and then integrating over $[0, t]$, we obtain

From (3.5), it is easy to verify

Assume that there exists a time $t_1\in (0, T)$ such that $u_m(t_1)\notin \mathcal{W}$, then, by the continuity of $u_m(t)$ on $t$, we get $u_m(t_1)\in\partial \mathcal{W}$. Thus, we receive either

or

From (3.5), we have $J(u_m(t_1)) < d$. Thus, the case (3.7) is impossible.

If (3.8) is valid, $u_m(t_1)\in\mathcal{ N}$. From (2.13), we obtain $J(u_m(t_1))\geq d$. This contradicts with (3.5). Therefore, the case (3.8) is also impossible as well.

We deduce from (2.10), (3.5) and (3.6) that

which implies that

Taking $a = \sqrt{\pi}$ in (2.2), we have from Lemma 2.5, Eqs (2.8) and (2.9) that

Here $C_M = 2pM+\frac{(p-2)p^2}2M+2pM\ln(p^2M).$ From (3.5), we have

For $u, v\in W_0^{1, p}(\Omega)$, by (1.4), we have $(\Delta_pu, v) = \int_\Omega|\nabla u|^{p-2}|\nabla u|\cdot|\nabla v|dx)$. Hence, from Hölder inequality and (3.11), we obtain

It follows from (3.10)–(3.13) that the following limitations are true.

Combining (3.15) and (3.16) with Lemma 2.2 yields

which implies

Let $\Omega_1 = \{x\in\Omega:\ |u_m(x, t)\leq1\}$ and $\Omega_2 = \{x\in\Omega:\ |u_m(x, t)\geq1\}$, then by means of direct calculation, we know from Lemma 2.1 and Eq (3.11)

where $\frac1p+\frac1{p'} = 1$ and $L_M = [(p-1)e]^{-p'}|\Omega|+\bigg(\frac{n-p}{p(p-1)}\bigg)^{p'}C^{\frac{np}{n-p}}C_M^{\frac{n}{n-p}}$. From Lemma 2.3, Eqs (3.19) and (3.20), we receive

Now, we prove $\chi = \Delta_pu$. For this reason, multiplying both sides of (3.1) by an arbitrary smooth function $\varphi(t)\in C^2[0, T]$ and integrating over $[0, T]$, we have

Taking the limitation of both sides of Eq (3.22) with $j$ fixed and $m\rightarrow \infty$, we get

By (3.23), we have

for every $\psi\in L^2(0, T; W_0^{1, p}(\Omega)), \ \ \psi'\in L^2(0, T; L^2(\Omega))$. In particular, Setting $\psi = u$ in (3.24), we obtain

On the other hand, multiplying both sides of (3.1) by $g_{jm}(t)$, summing on $j$ from $1$ to $m$ and integrating over $[0, T]$, we get

Taking the inferior limitation on both sides of (3.26) as $m\rightarrow \infty$, we have

We conclude from (3.25) and (3.27) that

By the monotonicity of operator $\Delta_p$, we have

We get from (3.28) and (3.29) that

Combining (3.29) with (3.30) yields that

Let $v = u-\lambda\omega$, then, by (3.31), we obtain

for any $\omega\in L^p(0, T; W_0^{1, p}(\Omega))$ and any real number $\lambda$.

As $\lambda > 0, \lambda\rightarrow0$, from (3.32) and the hemicontinuity of operator $\Delta_p$, we conclude that

Similarly, when $\lambda < 0, \lambda\rightarrow0$, we have

Thus, for all $\omega\in L^p(0, T; W_0^{1, p}(\Omega))$, we deduce from (3.33) and (3.34) that

which implies that $\chi = \Delta_pu$.

Next, we prove above solution $u(x, t)$ satisfies (1.2), i.e., $u(x, 0) = u_0(x), \ u_t(x, 0) = u_1(x)$.

We conclude from Eqs (3.15), (3.16) and Lemma 1.2 that $u(t): [0, T]\rightarrow L^2(\Omega)$ is continuous. Therefore, $u_m(0)\rightarrow u(0)$ weakly in $L^2(\Omega)$. According to (3.3), one has $u(0) = u_0$.

To prove $u_t(0) = u_1$, let $\xi(t)$ be a smooth function with $\xi(0) = 1, \xi(T) = 0$. Noting

For given $j$, as $m\rightarrow \infty$, we get in the distribution sense

in ${\mathcal{D'}}([0, T])$. On the other hand,

converges to

as $m\rightarrow \infty$. Therefore,

From (3.36) and (3.37), we have $(u_t(0), \omega_j) = (u_1, \omega_j)$. By the density of$\{\omega_j\}_{j = 1}^m$ in $L^2(\Omega)$, we get $u_t(0) = u_1$. This completes the proof of Theorem 3.1.

For the case of ${\mathcal K}(u_0)\geq0$ and ${\mathcal E}(0) = M\leq d$, the global existence result of solutions to the problem (1.1)–(1.3) reads as follows:

Theorem 3.2. Assume that $p$ fulfils (1.5). If $u_0\in W_0^{1, p}(\Omega), \ u_1\in L^2(\Omega)$ and ${\mathcal E}(0) = M\leq d, \ {\mathcal K}(u_0)\geq0$, then there exists a global weak solution $u(x, t)$ for the problem (1.1)–(1.3) which satisfies $u(x, t)\in L^\infty([0, +\infty); W_0^{1, p}(\Omega)), \ u_t(x, t)\in L^\infty([0, +\infty); L^2(\Omega))$.

Proof. For the case $\|\nabla u_0\|_p\not = 0$, let us suppose that $\rho_k = 1-\frac1k$ and $u_{0k} = \rho_k u_0, \ k\geq2$. The problem (1.1)–(1.3) can be written as follows:

From ${\mathcal K}(u_0)\geq0$ and Lemma 2.6, we have $\theta^* = \theta^*(u_0)\geq1$. Accordingly, we get ${\mathcal K}(u_{0k}) > 0$. By (2.3), we obtain

Therefore, we receive

which implies that $u_{0k}\in \mathcal{W}$.

For each $k$, by Theorem 3.1, there exists a global weak solution $u_k(t)$ of the problem (3.38) such that $u_k(t)\in L^\infty([0, +\infty); W_0^{1, p}(\Omega)), \ \ u_{kt}(t)\in L^\infty([0, +\infty); L^2(\Omega))$ and

for any $v\in W_0^{1, p}(\Omega)$.

In addition,

By using (3.41) and combining with the same argument as (3.6), we can prove $u_k(t)\in \mathcal{W}.$

For the case $\|\nabla u_0\|_p = 0$, we get ${\mathcal J}(u_0) = 0$ by ${\mathcal K}(u_0)\geq0$. Thus, we have ${\mathcal E}(0) = \frac12\|u_1\|^2+{\mathcal J}(u_0) = \frac12\|u_1\|^2 = M.$ Let $\rho_k = 1-\frac1k, \ u_{1k} = \rho_ku_1(x), \ k\geq2$, we consider the following problem

Noting

By Eq (3.43) and Theorem 3.1, there is a global weak solution $u_k(t)$ for the problem (3.42) such that $u_k(t)\in L^\infty(0, +\infty; W_0^{1, p}(\Omega)), \ u_{kt}(t)\in L^\infty(0, +\infty; L^2(\Omega))$ and $u_k(t)\in \mathcal{W}$ for each $k$.

The remainder of the proof for Theorem 3.2 is the same as those of Theorem 3.1. Here, we omit them.

4.   Nonexistence of global solutions

Lemma 4.1. ^[38,39] If nonnegative function $\Phi(t)\in C^2$ satisfies

for $\Phi(0) > 0, \ \Phi'(0) > 0$ and $\rho > 0$, then there exists a time $T_*$ such that $0 < T_*\leq \frac{\Phi(0)}{\rho \Phi'(0)}$ and $\lim_{t\rightarrow T_*^-}\Phi(t) = +\infty$.

Lemma 4.2. Suppose that $u(t)$ is a solution of (1.1)–(1.3). If $u_0\in{\mathcal{U}}$ and ${\mathcal E}(0) < d$, then $u(t)\in{\mathcal{U}}$ and ${\mathcal E}(t) < d, \ \forall t\geq0$.

Proof. From the conservation of energy, we obtain ${\mathcal E}(t) = {\mathcal E}(0) < d.$ From (2.11), we get

Assume that there is $t^*\in[0, +\infty)$ such that $u(t^*)\not\in{\mathcal{U}}$, then by continuity of ${\mathcal K}(u(t))$ on $t$, we obtain ${\mathcal K}(u(t^*)) = 0$. That means $u(t^*)\in{\mathcal{N}}$. From (2.14), we have ${\mathcal J}(u(t^*))\geq d$, which is contradiction with (4.1). Therefore, the conclusion in Lemma 4.2 holds.

Theorem 4.1. Suppose that $0 < {\mathcal E}(0) < d$ and $\int_\Omega u_0u_1dx > 0$, then there is no global weak solution $u(t)$ to the problem (1.1)–(1.3). Namely, there exists a time $T_*$ such that $\lim_{t\rightarrow T_*^-}\|u(t)\|^2 = +\infty$, where the lifespan $T_*$ is estimated by $0 < T_* < \frac{4\Psi(0)}{(p-2)\Psi'(0)}$, $\Psi(t)$ is given in (4.19).

Proof. By $u_0\in{\mathcal{U}}$, ${\mathcal E}(0) < d$ and Lemma 4.2, we get $u\in{\mathcal{U}}$. Thus,

From (2.13) and (2.19), we have

We deduce from (2.17), (4.2) and (4.3) that

Let

Then there is a real number $\alpha > 0$, which satisfies

By differentiating on both sides of (4.5), we get

From (4.7), we obtain

Combining (1.1) with (4.8), we get

By $u\in \mathcal{U}$ and (4.9), we receive $\Psi''(t) > 0$. Combining (4.5), (4.7) and (4.9), we get

where

By Cauchy-Schwarz inequality, we get

This inequality (4.12) guarantees $\Upsilon(t)\geq0$. By (4.10), we have

where

From (2.11) and (4.14), we obtain

By (4.4), (4.15) and ${\mathcal E}(t) = {\mathcal E}(0) < d$, we get

Therefore, there exists $\beta > 0$ such that

Combining (4.6), (4.13) and (4.17), we conclude that

Let $\rho = \frac{p-2}{4} > 0$, then, by the differential inequality (4.18) and Lemma 4.1, one has

and

From (4.5) and (4.20), we have $\lim_{t\rightarrow T_*^-}\|u(t)\|^2 = +\infty.$

This completes the proof of Theorem 4.1.

5.   Conclusions

By applying Galerkin method and $L^p$-Sobolev logarithmic inequality, and combining with the potential well theory, we prove the global existence result of solutions in this paper. Namely, assume that $p$ satisfies (1.5). If $u_0\in W_0^{1, p}(\Omega), \ u_1\in L^2(\Omega)$ and $0 < {\mathcal{E}}(0)\leq M, \ {\mathcal{K}}(u_0)\geq0$, then there is a global weak solution $u(x, t)$ of the problem (1.1)–(1.3). Meanwhile, under the condition of positive initial energy, by using the concavity analysis method, we establish the finite time blow-up result of solutions and give the lifespan estimate of solutions. The result read as follows: If $0 < {\mathcal E}(0) < d$ and $\int_\Omega u_0u_1dx > 0$, then the solutions of the problem (1.1)–(1.4) blows up in finite time and the lifespan $T_{*}$ is estimated by $0 < T_* < \frac{4\Psi(0)}{(p-2)\Psi'(0)}$.

Acknowledgments

The authors would like to thank the reviewers and editors for their help to improve the quality of this article. Moreover, this research was supported by Natural Science Foundation of Zhejiang Province (No. LY17A010009).

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.