Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

Abstract

In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we derive a growth estimate for the blowing up global solutions.

1. Introduction

In this paper, we study the initial boundary value problem for the nonlinear wave equation

$$u_{tt}-\Delta u=f(x,u),t>0,x\in \Omega ,$$

(1)

$$\left\{\begin{array}{cc}u(0,x)=u_0\left(x\right),\hfill & u_t(0,x)=u_1\left(x\right),x\in \Omega ,\hfill \\ u(t,x)=0,\hfill & t\ge 0,x\in \partial \Omega ,\hfill \\ u_0\left(x\right)\in {\mathrm{H}}_0^1(\Omega ),\hfill & u_1\left(x\right)\in {\mathrm{L}}^2(\Omega ).\hfill \end{array}\right.$$

(2)

Here, $\Omega$ is a bounded open subset of ${\mathbb{R}}^n$$(n\ge 1)$ with smooth boundary $\partial \Omega$. The nonlinearity $f(x,u)$ has the following form:

$$f(x,u)=\left\{\begin{array}{c}\begin{array}{cc}{\displaystyle a_0{\left(x\right)uln\left|u\right|}^k+\sum _{i=1}^r{a}_i\left(x\right){\left|u\right|}^{p_i-1}u}\hfill & \mathrm{if}\left|u\right|>0;\hfill \\ 0\hfill & \mathrm{if}u=0,\hfill \end{array}\hfill \end{array}\right.$$

(3)

where k, the functions $a_i\left(x\right)$, $i=0,...,r$, and the power exponents $p_i$, $i=1,...,r$ satisfy the following conditions:

$$\begin{array}{cc}\hfill & k>0,a_0\left(x\right)\in \mathrm{C}\left(\overline{\Omega }\right),a_0\left(x\right)\ge a>0\forall x\in \overline{\Omega },a=\mathrm{const},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{a}_i\left(x\right)\in \mathrm{C}\left(\overline{\Omega }\right),0\le a_i\left(x\right)\le A_i,\forall x\in \overline{\Omega },A_i=\mathrm{const}>0,i=1,\dots ,r,\hfill \\ 1<p_1<\cdots <p_r,\hfill \\ {\displaystyle p_r<\infty \mathrm{if}n=1,2;p_r\le \frac{n+2}{n-2}\mathrm{if}n\ge 3.}\hfill \end{array}\right.\hfill \end{array}$$

(5)

Problems with logarithmic nonlinearities have attracted a lot of attention in recent years. The real applications of logarithmic nonlinearities appear via different type of equations, such as the logarithmic Klein–Gordon equation in nuclear physics, inflation cosmology, vibration, quantum field theory, spinless particles, and viscoelastic mechanics (see, e.g., [1,2,3,4,5,6]); the logarithmic Schrodinger equation in quantum optics and transport phenomena (see, e.g., [1,7]); and the logarithmic wave equation in fluid dynamics (see, e.g., [8]).

One of the powerful methods for studying the solutions to nonlinear evolution equations is the potential well method, introduced in [9]. For sub-critical initial energy $0<E\left(0\right)<d$, this method gives a complete characterization of the solutions depending on the sign of the Nehari functional at the initial time $I\left(u_0\right)$ (see the definitions of $E\left(0\right)$, d, and $I\left(u\right(t\left)\right)$ in (15), (11), and (10), respectively). The potential well method, developed in the pioneering paper [9], has been successfully applied in the investigation of undamped wave and Klein–Gordon equations with a wide variety of Lipschitz nonlinearities. For example, nonlinearities of the type $f\left(u\right)={au\left|u\right|}^{p-1},f\left(u\right)=a{\left|u\right|}^p,p>1$, $a=\mathrm{const}>0$ are studied, e.g., in [9,10,11]. More general cases of combined power-type nonlinearities with constant coefficients are considered in [12,13], while combined power-type nonlinearities with variable coefficients are treated in [14,15]. For Lipschitz nonlinearities of the type $f\left(u\right)=u^{p}ln\left|u\right|$, $p>1$, we refer to [16]. The case of wave and Klein–Gordon equations with damping terms and polynomial-type nonlinearities is also treated by the potential well method; see, e.g., [17,18,19].

The first results of the study of the Cauchy problem to the Klein–Gordon equation with locally non-Lipschitz nonlinearity,

$$f\left(u\right)=kuln\left|u\right|,k>0$$

(6)

are given in [20], where the authors prove the existence and uniqueness of the solutions in ${\mathbb{R}}^3$. Another approach for proving the existence of a result for the initial boundary value problem to the Klein–Gordon equation with nonlinearity (6) in $\Omega =[a,b]\subset \mathbb{R}$ is developed in [21]; see also [22] for $\Omega \subset {\mathbb{R}}^n$. Later on, the initial boundary value problem (1) and (2) with nonlinearity (6) is considered in [23]. Using the potential well method, the global existence and infinite-time blow up of the weak solutions have been proved when $0<E\left(0\right)\le d$. Analogous results for the Klein–Gordon equation with nonlinearity (6) are obtained in [24]. In the case of arbitrary positive initial energy, a sufficient condition for blow up at infinity of the solutions to problem (1) and (2), (6) is given in [23]. Moreover, the growth of the blowing up solutions at infinity is derived in [25].

In the last decade, Klein–Gordon equations with a linear dumping term

$$u_{tt}-\Delta u+u+u_t=f\left(u\right),$$

(7)

have been intensively investigated for locally non-Lipschitz nonlinearities. Equation (7) with nonlinearity $f\left(u\right)$, defined by (6), is studied in [26,27,28]. When $f\left(u\right)={uln\left|u\right|}^2-u{\left|u\right|}^2$, the global existence of the solutions to (7) is obtained in [29]. Let us also note the result in [30], where infinite time blow up of the solutions to (7) with nonlinearity (6) is proved for arbitrarily high energy.

The initial boundary value problem for the nonlinear wave equation

$$u_{tt}-\Delta u+\omega \nabla u_t+\mu u_t=uln\left|u\right|,\omega \ge 0,\mu >0$$

is considered in [31]. In the framework of the potential well method, the global existence, energy decay, and infinite time blow up of the solution with $0<E\left(0\right)\le d$ are proved. Moreover, for high energy, the infinite time blow up result is established.

Finally, let us note that logarithmic nonlinearities are investigated for different types of equations such as delay Klein–Gordon wave-type equations [32], Boussinesq-type equations [33], plate equations [34,35], p-Kirchhoff problems [36], parabolic equations [37,38], etc.

The aim of this paper is to investigate the infinite time blow up solutions to problem (1) and (2) with nonlinearity (3)–(5). This nonlinearity includes a locally non-Lipschitz logarithmic term as well as power-type nonlinear terms. Moreover, the coefficients of the nonlinearity are non-negative functions of the space variables. It is important to point out that nonlinearity (3)–(5) has not been studied so far.

For the first time, a necessary and sufficient condition for infinite time blow up of the global solutions to (1)–(5) is proved for arbitrary initial energy $E\left(0\right)$; see Theorem 1. The necessary and sufficient condition (25) is formulated at some time $b\ge 0$ and it is independent of the sign of the scalar product $(u\left(b\right),u_t\left(b\right))$. As a consequence, we obtain a new sufficient condition for blow up at infinity of the global solutions; see (32) in Proposition 1. Note, that in contrast to [23], this sufficient condition does not impose any requirements on the sign of the scalar product $(u_0,u_1)$ and the sign of $I\left(u_0\right)$. Furthermore, we derive a growth estimate for the infinite time blow up solutions; see Theorem 2. For nonlinearity (6), which is a special case of (3), we obtain a faster estimate than the estimate given in [25].

It is important to emphasize that all of the results presented in this paper are also valid for problem (1) and (2) with the following more general nonlinearities:

$$f(x,u)=a_0{\left(x\right)uln\left|u\right|}^k+f_1(x,u)\mathrm{if}\left|u\right|>0;f(x,0)=0,$$

(8)

where k and $a_0\left(x\right)$ satisfy conditions (4), while for the function $f_1(x,w)$ we impose the requirements

$$\left\{\begin{array}{c}{f}_1(x,0)=0;\hfill \\ f_1(x,w)\mathrm{is}\mathrm{a}\mathrm{continuous}\mathrm{function}\mathrm{in}x\in \overline{\Omega };\hfill \\ f_1(x,w)\mathrm{is}\mathrm{a}\mathrm{locally}\mathrm{Lipschitz}-\mathrm{continuous}\mathrm{function}\mathrm{in}w\in \mathbb{R};\hfill \\ {\displaystyle w{f}_1(x,w)-2{\int }_0^w{f}_1(x,s)ds\ge 0,\forall x\in \overline{\Omega },\forall w\in \mathbb{R}.}\hfill \end{array}\right.$$

(9)

Note that when $f_1(x,u)\equiv -u$, Equation (1) with nonlinearity (8) becomes the Klein–Gordon equation with logarithmic nonlinearity. The arguments proving the possibility of replacing nonlinearity (3)–(5) by nonlinearity (8), (4), (9) are given in Remark 1.

The paper is organized as follows. Some preliminary notes and definitions are given in Section 2. Section 3 deals with auxiliary ordinary differential equations. The main results are formulated and proved in Section 4. In Section 5, for a special case of nonlinearity, the new sufficient condition for infinite time blow up and the growth estimate are discussed and compared with previous ones.

2. Preliminary

Throughout the paper we use the following short notation for the functions depending on t and x:

$$\begin{array}{c}\hfill {\parallel u\left(t\right)\parallel }_p={\parallel u(t,·)\parallel }_{{\mathrm{L}}^p(\Omega )},1<p\le \infty ,(u\left(t\right),v\left(t\right))={\int }_{\Omega }u(t,x)v(t,x)dx\end{array}$$

and for convenience we write $\parallel ·\parallel$ instead of ${\parallel ·\parallel }_2$.

Definition 1.

The function $u(t,x)$ is a weak solution to problem (1)–(5) if 

$$u(t,x)\in \mathrm{C}((0,T_m);{\mathrm{H}}_0^1(\Omega ))\cap {\mathrm{C}}^1((0,T_m);{\mathrm{L}}^2(\Omega ))\cap {\mathrm{C}}^2((0,T_m);{\mathrm{H}}^{-1}(\Omega ))$$

and the identity 

$$\begin{array}{c}{\int }_{\Omega }{u}_t(t,x)\eta \left(x\right)dx+{\int }_0^t{\int }_{\Omega }\nabla u(\tau ,x)\nabla \eta \left(x\right)dxd\tau \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\int }_0^t{\int }_{\Omega }f(x,u(\tau ,x))\eta \left(x\right)dxd\tau +{\int }_{\Omega }{u}_1\left(x\right)\eta \left(x\right)dx\hfill \end{array}$$

holds for every $\eta \left(x\right)\in {\mathrm{H}}_0^1(\Omega )$ and every $t\in [0,T_m)$.

Definition 2.

Suppose $u(t,x)$ is a weak solution to problem (1)–(5) in the maximal existence time interval $[0,T_m)$, $0<T_m\le \infty$. Then, solution $u(t,x)$ blows up at $T_m$ if

$$\underset{t\to T_m,t<T_m}{lim\; sup}\parallel u\left(t\right)\parallel =\infty .$$

Let us give some definitions that play an important role in studying the behavior of the solutions. For every $w\left(x\right)\in {\mathrm{H}}_0^1(\Omega )$, we introduce the potential energy functional $J\left(w\right)$,

$$\begin{array}{cc}\hfill J\left(w\right):=& {\displaystyle \frac{1}{2}}{\parallel \nabla w\parallel }^2-{\int }_{\Omega }{\int }_0^{w\left(x\right)}f(x,z)dzdx\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\parallel \nabla w\parallel }^2-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{a}_0\left(x\right)w^2{ln\left|w\right|}^{k}dx+{\displaystyle \frac{k}{4}}{\int }_{\Omega }{a}_0\left(x\right)w^{2}dx-\sum _{i=1}^r{\int }_{\Omega }{\displaystyle \frac{a_i\left(x\right)}{p_i+1}}{\left|w\right|}^{p_i+1}dx,\hfill \end{array}$$

the Nehari functional $I\left(w\right)$,

$$\begin{array}{cc}\hfill I\left(w\right):=& {\parallel \nabla w\parallel }^2-{\int }_{\Omega }wf(x,w)dx\hfill \\ \hfill =& {\parallel \nabla w\parallel }^2-{\int }_{\Omega }{a}_0\left(x\right)w^2{ln\left|w\right|}^{k}dx-\sum _{i=1}^r{\int }_{\Omega }{a}_i\left(x\right){\left|w\right|}^{p_i+1}dx\hfill \end{array}$$

(10)

and the critical energy constant d (the depth of the potential well),

$$d=\underset{w\in \mathcal{N}}{inf}J\left(w\right),$$

(11)

where the Nehari manifold $\mathcal{N}$ is defined by

$$\mathcal{N}=\{w\in {\mathrm{H}}_0^1(\Omega ):I\left(w\right)=0,\parallel w\parallel \ne 0\}.$$

Straightforward computations give us the following relation between the functionals $I\left(w\right)$ and $J\left(w\right)$:

$$J\left(w\right)={\displaystyle \frac{1}{2}}I\left(w\right)+{\displaystyle \frac{ka}{4}}{\parallel w\parallel }^2+B\left(w\right),$$

(12)

where

$$\begin{array}{c}\hfill B\left(w\right)={\displaystyle \frac{k}{4}}{\int }_{\Omega }(a_0\left(x\right)-a){\left|w\right|}^{2}dx+\sum _{i=1}^r{\int }_{\Omega }{\displaystyle \frac{(p_i-1)}{2(p_i+1)}}{a}_i\left(x\right){\left|w\right|}^{p_i+1}dx.\end{array}$$

(13)

Due to the conditions (4), (5) on the functions $a_i\left(x\right)$ and power exponents $p_i$ in $f(x,u)$, we conclude that

$$B\left(w\right)\ge 0,\mathrm{for}\mathrm{all}w\in {\mathrm{H}}_0^1(\Omega ).$$

(14)

When the functionals I, J, and B are evaluated on functions depending on t and x, then we use the short notation $I\left(t\right)=I\left(u\right(t\left)\right)=I\left(u\right(t,·\left)\right)$, $J\left(t\right)=J\left(u\right(t\left)\right)=J\left(u\right(t,·\left)\right)$, $B\left(t\right)=B\left(u\right(t\left)\right)=B\left(u\right(t,·\left)\right)$.

If $u(t,x)$ is a weak solution to problem (1)–(5) in the maximal existence time interval $[0,T_m)$, $0<T_m\le \infty$, we define the energy functional $E\left(t\right)$ as

$$\begin{array}{cc}\hfill E\left(t\right)=E\left(u\right(t\left)\right):& ={\displaystyle \frac{1}{2}}\parallel u_t{\left(t\right)\parallel }^2+{\displaystyle \frac{1}{2}}{\parallel \nabla u\left(t\right)\parallel }^2-{\int }_{\Omega }{\int }_0^{u(t,x)}f(x,z)dzdx\hfill \\ & ={\displaystyle \frac{1}{2}}\parallel u_t{\left(t\right)\parallel }^2+{\displaystyle \frac{1}{2}}{\parallel \nabla u\left(t\right)\parallel }^2-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{a}_0\left(x\right)u^2(t,x)ln{\left|u(t,x)\right|}^{k}dx\hfill \\ & +{\displaystyle \frac{k}{4}}{\int }_{\Omega }{a}_0\left(x\right)u^2(t,x)dx-\sum _{i=1}^r{\int }_{\Omega }{\displaystyle \frac{a_i\left(x\right)}{p_i+1}}{\left|u(t,x)\right|}^{p_i+1}dx.\hfill \end{array}$$

(15)

Note that $u(t,x)$ satisfies the conservation law

$$E\left(0\right)=E\left(t\right),\forall t\in [0,T_m).$$

(16)

Indeed, differentiating (15) with respect to t, we obtain from (1) the identity

$$E^{\prime }\left(t\right)={\int }_{\Omega }{u}_t(t,x)(u_{tt}(t,x)-\Delta u(t,x)-f(x,u(t,x)))dx=0\mathrm{for}t\in [0,T_m),$$

which proves (16).

Moreover, definition (15), the conservation law (16), and (10) give us the following helpful relation:

$$I\left(u\left(t\right)\right)=2E\left(0\right)-\parallel u_t{\left(t\right)\parallel }^2-ka{\parallel u\left(t\right)\parallel }^2-2B\left(u\left(t\right)\right).$$

(17)

3. Basic Ordinary Differential Equations

The proof of the main results of the paper is based on the study of the behavior of the non-negative smooth function $\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2$. Below, we recall some preliminary results.

Definition 3.

We say that a non-negative function $\psi \left(t\right)\in {\mathrm{C}}^2\left([0,T_m)\right)$, $0<T_m\le \infty$ blows up at $T_m$ if

$$\underset{t\to T_m,t<T_m}{lim\; sup}\psi \left(t\right)=\infty .$$

Lemma 1

(Lemma 2.2 in [39]). Suppose $\psi \left(t\right)\in C^1\left([0,T_m)\right)$, $0<T_m\le \infty$, is a non-negative function, and M is an arbitrary non-negative constant. If $\psi \left(t\right)$ blows up at $T_m$, then there exists $t_0$, $t_0\in [0,T_m)$ such that $\psi \left(t_0\right)\ge M$ and ${\psi }^{\prime }\left(t_0\right)>0$.

Lemma 2.

If $u(t,x)$ is a global weak solution to problem (1)–(5), then $\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2$ is a solution of the equation

$${\psi }^{″}\left(t\right)=\alpha \psi \left(t\right)-\beta +H\left(t\right),$$

(18)

where $\alpha =ka$, $\beta =4E\left(0\right)$,

$$H\left(t\right)=4(\parallel u_t\left(t\right){\parallel }^2+B\left(u\left(t\right)\right))\ge 0\forall t\ge 0$$

(19)

and $B\left(u\right(t\left)\right)$ is defined by (13).

Moreover, the classical solution of (18) for $t\in [b,\infty )$, $b\ge 0$ is given by the formula

$$\begin{array}{cc}\hfill \psi \left(t\right)=& {\displaystyle \frac{1}{2}}\left(\psi \left(b\right)+{\displaystyle \frac{1}{\sqrt{\alpha }}}{\psi }^{\prime }\left(b\right)-{\displaystyle \frac{\beta }{\alpha }}\right)e^{\sqrt{\alpha }(t-b)}+{\displaystyle \frac{1}{2}}\left(\psi \left(b\right)-{\displaystyle \frac{1}{\sqrt{\alpha }}}{\psi }^{\prime }\left(b\right)-{\displaystyle \frac{\beta }{\alpha }}\right)e^{-\sqrt{\alpha }(t-b)}\hfill \\ & +{\displaystyle \frac{\beta }{\alpha }}+{\displaystyle \frac{1}{\sqrt{\alpha }}}{\int }_b^{t}H\left(s\right)sinh\left(\sqrt{\alpha }(t-s)\right)ds.\hfill \end{array}$$

(20)

Proof. 

Direct computations give us the following formulas for the derivatives of the function $\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2$:

$${\psi }^{\prime }\left(t\right)=2(u\left(t\right),u_t\left(t\right)),{\psi }^{″}\left(t\right)=2{\parallel u_t\left(t\right)\parallel }^2+2(u\left(t\right),u_{tt}\left(t\right)).$$

Since $u(t,x)$ is a weak solution of (1), we have

$$\begin{array}{cc}\hfill {\psi }^{″}\left(t\right)=& 2\parallel u_t{\left(t\right)\parallel }^2+2(u\left(t\right),\nabla u\left(t\right)+f(x,u(t,x)))\hfill \\ \hfill & =2\parallel u_t{\left(t\right)\parallel }^2-{2\parallel \nabla u\left(t\right)\parallel }^2+2(u\left(t\right),f(x,u(t,x)))=2{\parallel u_t\left(t\right)\parallel }^2-2I\left(u\left(t\right)\right).\hfill \end{array}$$

Using (17), we obtain

$$\begin{array}{c}\hfill {\psi }^{″}\left(t\right)=2\parallel u_t{\left(t\right)\parallel }^2-2I\left(u\left(t\right)\right)=4\parallel u_t{\left(t\right)\parallel }^2-4E\left(0\right)+ka{\parallel u\left(t\right)\parallel }^2+4B\left(u\left(t\right)\right).\end{array}$$

(21)

Note that Equation (21) is equivalent to (18) with $\alpha =ka$, $\beta =4E\left(0\right)$, and $H\left(t\right)$ defined in (19). From (14) it follows that $H\left(t\right)\ge 0$ for every $t\ge 0$. Finally, we obtain Formula (20) calculating (18) by means of the variation of the constants for $t\in [b,\infty )$. Lemma 2 is proved. □

Lemma 3.

Suppose $u(t,x)$ is a global weak solution to problem (1)–(5) and there exists a time $t_{\ast }\ge 0$ such that $\parallel u\left(t\right)\parallel >0$ for every $t\ge t_{\ast }$. Then, $\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2$ is a solution of the equation

$${\psi }^{″}\left(t\right)={\displaystyle \frac{{\psi }^{\prime 2}\left(t\right)}{\psi \left(t\right)}}+\alpha \psi \left(t\right)-\beta +Q\left(t\right)\forall t\ge t_{\ast },$$

(22)

where $\alpha =ka$, $\beta =4E\left(0\right)$ and

$$Q\left(t\right)=4(B\left(u\left(t\right)\right)+\left(\parallel u_t{\left(t\right)\parallel }^2-{\displaystyle \frac{{(u\left(t\right),u_t\left(t\right))}^2}{{\parallel u\left(t\right)\parallel }^2}}\right))\ge 0\forall t\ge t_{\ast }.$$

(23)

Proof. 

Since $\parallel u\left(t\right)\parallel >0$, $t\ge t_{\ast }$, we consider the following decomposition:

$$u_t(t,x)=h(t,x)+{\displaystyle \frac{(u\left(t\right),u_t\left(t\right))}{{\parallel u\left(t\right)\parallel }^2}}u(t,x),\mathrm{where}(h\left(t\right),u\left(t\right))=0.$$

Thus,

$${\parallel u_t\left(t\right)\parallel }^2={\displaystyle \frac{{(u\left(t\right),u_t\left(t\right))}^2}{{\parallel u\left(t\right)\parallel }^2}}+{\parallel h\left(t\right)\parallel }^2={\displaystyle \frac{{\psi }^{\prime 2}\left(t\right)}{4\psi \left(t\right)}}+{\parallel h\left(t\right)\parallel }^2.$$

(24)

From (21) and expression (24), we obtain that $\psi \left(t\right)$ satisfies Equation (22) with $\alpha =ka$ and $\beta =4E\left(0\right)$, and

$$Q\left(t\right)=4(B\left(u\left(t\right)\right)+\parallel h\left(t\right){\parallel }^2).$$

In view of (14), we have $Q\left(t\right)\ge 0$ for every $t\ge t_{\ast }$, which completes the proof. □

4. Main Results

Below, we formulate and prove a necessary and sufficient condition for blow up at infinity of the global solutions to problem (1)–(5). Moreover, the growth estimate at infinity of the blowing up solutions is given.

Theorem 1

(Necessary and sufficient condition). If $u(t,x)$ is a global weak solution to problem (1)–(5), then:

(i) 

$u(t,x)$ blows up at infinity if

$$\mathit{there}\mathit{exists}b\ge 0\mathit{such}\mathit{that}E\left(0\right)<{\displaystyle \frac{ka}{4}}{\parallel u\left(b\right)\parallel }^2+{\displaystyle \frac{\sqrt{ka}}{2}}(u\left(b\right),u_t\left(b\right));$$

(25)

(ii) 

if (25) fails, then the estimate

$${\parallel u\left(t\right)\parallel }^2\le \left(\parallel u_0{\parallel }^2-{\displaystyle \frac{4E\left(0\right)}{ka}}\right)e^{-\sqrt{ka}t}+{\displaystyle \frac{4E\left(0\right)}{ka}}$$

holds for every $t\ge 0$.

Proof. 

(i) Sufficiency.

Suppose (25) holds. From Lemma 2 it follows that the function $\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2$ satisfies (18). Note that (25) is equivalent to the inequality

$$\psi \left(b\right)+{\displaystyle \frac{1}{\sqrt{ka}}}{\psi }^{\prime }\left(b\right)-{\displaystyle \frac{4E\left(0\right)}{ka}}=\psi \left(b\right)+{\displaystyle \frac{1}{\sqrt{\alpha }}}{\psi }^{\prime }\left(b\right)-{\displaystyle \frac{\beta }{\alpha }}>0.$$

Since $H\left(t\right)\ge 0$ for every $t\ge 0$, from formula (20) we obtain $\psi \left(t\right)\to \infty$ for $t\to \infty$. i.e., $u(t,x)$ blows up at infinity.

(i) Necessity. Suppose $u(t,x)$ blows up at infinity. If $E\left(0\right)>0$, then from Lemma 1 for $T_m=\infty$ and $M=4E\left(0\right)/\left(ka\right)$ it follows that there exists $t_0\in [0,T_m)$ such that ${\Psi }^{\prime }\left(t_0\right)>0$ and $\Psi \left(t_0\right)\ge 4E\left(0\right)/\left(ka\right)$. Hence, condition (25) is satisfied for $b=t_0$. In case $E\left(0\right)\le 0$, the proof is analogous to the previous case, the difference being that we apply Lemma 1 with $M=0$.

(ii) Since condition (25) fails, then inequality

$$\psi \left(t\right)+{\displaystyle \frac{1}{\sqrt{ka}}}{\psi }^{\prime }\left(t\right)-{\displaystyle \frac{4E\left(0\right)}{ka}}\le 0$$

(26)

holds for every $t\ge 0$. Integrating (26), we have for every $t\ge 0$ the estimate

$$\begin{array}{cc}\hfill \psi \left(t\right)& \le e^{-\sqrt{ka}t}\left(\psi \left(0\right)+{\displaystyle \frac{4E\left(0\right)}{\sqrt{ka}}}{\int }_0^t{e}^{\sqrt{ka}s}ds\right)\hfill \\ \hfill =& \left(\psi \left(0\right)-{\displaystyle \frac{4E\left(0\right)}{ka}}\right)e^{-\sqrt{ka}t}+{\displaystyle \frac{4E\left(0\right)}{ka}}.\hfill \end{array}$$

Theorem 1 is proved. □

Theorem 2.

Suppose $u(t,x)$ is a global weak solution to problem (1)–(5) that blows up at infinity.

(i) 

If $E\left(0\right)>0$, then for every $\epsilon \in (0,ka)$ there exists a sufficiently large time $t_0=t_0\left(\epsilon \right)$ such that

$${\epsilon \parallel u\left(t\right)\parallel }^2>4E\left(0\right)\mathit{and}(u\left(t\right),u_t\left(t\right))>0\mathit{for}t>t_0.$$

(27)

Moreover, the estimate

$${\parallel u\left(t\right)\parallel }^2\ge {\parallel u\left(t_0\right)\parallel }^{2}exp\left[{\displaystyle \frac{2(u\left(t_0\right),u_t\left(t_0\right))}{\parallel u\left(t_0\right){\parallel }^2}}(t-t_0)+{\displaystyle \frac{ka-\epsilon }{2}}{(t-t_0)}^2\right]$$

(28)

holds for $t\ge t_0$.

(ii) 

If $E\left(0\right)=0$, then there exists a sufficiently large time $t_0$ such that

$${\parallel u\left(t\right)\parallel }^2>0\mathit{and}(u\left(t\right),u_t\left(t\right))>0\mathit{for}t>t_0.$$

Moreover, the estimate

$${\parallel u\left(t\right)\parallel }^2\ge {\parallel u\left(t_0\right)\parallel }^{2}exp\left[{\displaystyle \frac{2(u\left(t_0\right),u_t\left(t_0\right))}{\parallel u\left(t_0\right){\parallel }^2}}(t-t_0)+{\displaystyle \frac{ka}{2}}{(t-t_0)}^2\right]$$

(29)

holds for $t\ge t_0$.

Proof. 

(i) Since $u(t,x)$ blows up at infinity, for $E\left(0\right)>0$ we apply Lemma 1 with $T_m=\infty$ and $M=4E\left(0\right)/\epsilon$, where $\epsilon \in (0,ka)$. This guarantees that there exists $t_0\ge 0$ such that ${\psi }^{\prime }\left(t_0\right)>0$ and $\psi \left(t_0\right)\ge 4E\left(0\right)/\epsilon$. Now, we will show that the inequalities in (27) hold for every $t>t_0$, i.e., ${\psi }^{\prime }\left(t\right)>0$ and $\psi \left(t\right)>4E\left(0\right)/\epsilon$ for $t>t_0$.

From Equations (22) and (23), we have for $t=t_0$ the following inequalities:

$${\psi }^{″}\left(t_0\right)\ge {\displaystyle \frac{{\psi }^{\prime 2}\left(t_0\right)}{\psi \left(t_0\right)}}+ka\psi \left(t_0\right)-4E\left(0\right)>0,$$

which give that ${\psi }^{\prime }\left(t\right)>{\psi }^{\prime }\left(t_0\right)>0$ for $t\in [t_0,t_0+ϵ)$ for some sufficiently small $ϵ>0$. Suppose, by contradiction, that there exists an interval $(t_0,t_1)$, $t_1>t_0$ such that ${\psi }^{\prime }\left(t\right)>0$ for $t\in [t_0,t_1)$ and ${\psi }^{\prime }\left(t_1\right)=0$. Since $\psi \left(t\right)$ is a strictly monotone increasing function for $t\in [t_0,t_1]$, it follows that $\psi \left(t\right)>\psi \left(t_0\right)\ge 4E\left(0\right)/\epsilon >0$ for every $t\in (t_0,t_1]$. Moreover, from (22) and (23) we have that

$${\psi }^{″}\left(t\right)\ge {\displaystyle \frac{{\psi }^{\prime 2}\left(t\right)}{\psi \left(t\right)}}+(ka-\epsilon )\psi \left(t\right)+\epsilon \psi \left(t\right)-4E\left(0\right)>\epsilon \psi \left(t_0\right)-4E\left(0\right)\ge 0$$

for every $t\in (t_0,t_1]$. Hence, ${\psi }^{\prime }\left(t\right)$ is a strictly increasing function for $t\in (t_0,t_1]$ and we obtain the following impossible chain of inequalities:

$$0={\psi }^{\prime }\left(t_1\right)>{\psi }^{\prime }\left(t_0\right)>0.$$

Therefore, ${\psi }^{\prime }\left(t\right)>0$ for every $t>t_0$ and $\psi \left(t\right)>\psi \left(t_0\right)\ge 4E\left(0\right)/\epsilon$ for every $t>t_0$; i.e., (27) is satisfied.

Since $\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2>0$ for $t\ge t_0$ we apply Lemma 3 with $t_{\ast }=t_0$. So, (22), (23), and the inequalities in (27) give us

$${\psi }^{″}\left(t\right)\ge {\displaystyle \frac{{\psi }^{\prime 2}\left(t\right)}{\psi \left(t\right)}}+(ka-\epsilon )\psi \left(t\right)\mathrm{for}t\ge t_0,$$

or equivalently,

$$\psi \left(t\right){(ln\psi \left(t\right))}^{″}\ge (ka-\epsilon )\psi \left(t\right)\mathrm{for}t\ge t_0.$$

(30)

Integrating (30) twice from $t_0$ to t, we obtain

$${(ln\psi \left(t\right))}^{\prime }-{(ln\psi \left(t_0\right))}^{\prime }\ge (ka-\epsilon )(t-t_0),$$

$$ln\psi \left(t\right)\ge ln\psi \left(t_0\right)+{(ln\psi \left(t_0\right))}^{\prime }(t-t_0)+{\displaystyle \frac{(ka-\epsilon )}{2}}{(t-t_0)}^2.$$

Thus, (28) is proved.

(ii) The proof for $E\left(0\right)=0$ is analogous to the proof in (i). In this case, we apply Lemma 1 with $M=0$ and obtain

$${\psi }^{″}\left(t\right)\ge {\displaystyle \frac{{\psi }^{\prime 2}\left(t\right)}{\psi \left(t\right)}}+ka\psi \left(t\right)\mathrm{for}t\ge t_0.$$

(31)

Integrating (31) twice from $t_0$ to t, we obtain the estimate (29). The proof of Theorem 2 is completed. □

Theorem 3.

If $u(t,x)$ is a global weak solution to problem (1)–(5) and $E\left(0\right)<0$, then $u(t,x)$ blows up at infinity. Moreover, there exists a sufficiently large time $t_0$ such that $(u\left(t\right),u_t\left(t\right))>0$ and the estimate (29) holds for every $t>t_0$.

Proof. 

If $E\left(0\right)<0$, from (21) and (14) it follows that

$${\psi }^{″}\left(t\right)\ge -4E\left(0\right)>0\mathrm{for}t\ge 0,$$

which yields that $\psi \left(t\right)$ blows up at infinity. The rest of the proof is the same as in the case $E\left(0\right)=0$ in Theorem 2. □

As a consequence of Theorems 1 and 2, we obtain the following new sufficient condition for blow up of the solutions to problem (1)–(5) without any restrictions on the sign of the scalar product $(u_0,u_1)$.

Proposition 1

(Sufficient condition). Suppose $u(t,x)$ is a global weak solution to problem (1)–(5) and $E\left(0\right)>0$. If

$$E\left(0\right)<{\displaystyle \frac{ka}{4}}{\parallel u_0\parallel }^2+{\displaystyle \frac{\sqrt{ka}}{2}}(u_0,u_1),$$

(32)

then $u(t,x)$ blows up at infinity. Moreover, for every $t>t_0$ estimate (28) holds, where $t_0$ is a sufficiently large time such that the inequalities in (27) hold with $\epsilon \in (0,ka)$.

Recall, that in the framework of the potential well method the sign of the Nehari functional at the initial moment $I\left(u_0\right)$ is crucial for the behavior of the solution for problems with sub-critical initial energy $0<E\left(0\right)<d$. More precisely, if $0<E\left(0\right)<d$ and $I\left(u_0\right)<0$, then the weak solution blows up.

In the following statement, we show that the sufficient condition (32) guarantees a negative sign of the Nehari functional for any positive initial energy.

Proposition 2.

If the initial data $u_0$, $u_1$ for problem (1)–(5) satisfy (32), then $I\left(u_0\right)<0$.

Proof. 

Assume, by contradiction, that $I\left(u_0\right)\ge 0$. From (17) and (14), it follows that

$$E\left(0\right)\ge {\displaystyle \frac{1}{2}}\parallel u_1{\parallel }^2+{\displaystyle \frac{ka}{2}}{\parallel u_0\parallel }^2.$$

(33)

From (33) and (32), we deduce

$${\displaystyle \frac{1}{2}}\parallel u_1{\parallel }^2+{\displaystyle \frac{ka}{2}}\parallel u_0{\parallel }^2\le E\left(0\right)<{\displaystyle \frac{ka}{4}}{\parallel u_0\parallel }^2+{\displaystyle \frac{\sqrt{ka}}{2}}(u_0,u_1).$$

Combining the left-hand side and the right-hand side of the previous inequality, we obtain the following impossible inequality

$$\parallel \sqrt{ka}{u}_0-u_1{\parallel }^2+{\parallel u_1\parallel }^2<0.$$

Therefore, our assumption is incorrect and Proposition 2 is proved. □

Remark 1.

The detailed analysis of the proofs of the statements in this section shows that all the results are also valid for the global solutions of problem (1) and (2) with more general nonlinearity (8). Indeed, for nonlinearity (8) the functional $B\left(w\right)$, defined in (12), has the following form:

$$B\left(w\right)={\displaystyle \frac{k}{4}}{\int }_{\Omega }(a_0\left(x\right)-a){\left|w\right|}^{2}dx+{\displaystyle \frac{1}{2}}{\int }_{\Omega }\left(w{f}_1(x,w)-2{\int }_0^w{f}_1(x,s)ds\right)dx.$$

Under conditions (4) and (9), we obtain that $B\left(w\right)\ge 0$$\forall w\in {\mathrm{H}}_0^1(\Omega )$, which is crucial for the proofs of Lemmas 2 and 3, and consequently, for the proofs of Theorems 1–3 and Proposition 1.

5. Comparisons and Discussion

In this section, we compare our results with previous ones. Since nonlinearity (3)–(5) has not been studied so far, we consider a special case of (3)–(5) when

$$a_0\left(x\right)\equiv 1\mathrm{and}\sum _{i=1}^r\left|a_i\left(x\right)\right|\equiv 0\mathrm{in}\Omega .$$

So, we comment on the behavior of the solution to problem (1) and (2) with the following nonlinearity:

$$f(x,u)={uln\left|u\right|}^k,k>0\mathrm{if}\left|u\right|>0;f(x,0)=0.$$

(34)

5.1. Sub-Critical Case: $0<E\left(0\right)<d$

If the initial energy is sub-critical, i.e., $0<E\left(0\right)<d$, we demonstrate that the blow up result from Proposition 1 does not contradict the blow up result obtained by means of the potential well method. First, we recall the results for the global existence and infinite time blow up of the weak solutions to problem (1) and (2) with nonlinearity (34) obtained in [23] by the potential well method. For this, we introduce two subsets of ${\mathrm{H}}_0^1(\Omega )$ that play an important role in the framework of the potential well method:

$$W=\left\{z\in {\mathrm{H}}_0^1(\Omega ):I\left(z\right)>0\right\}\cup \left\{0\right\},V=\left\{z\in {\mathrm{H}}_0^1(\Omega ):I\left(z\right)<0\right\}.$$

According to Theorem 3.1 in [23], if $0<E\left(0\right)<d$ and $u_0\left(x\right)\in W$, then problem (1) and (2) with nonlinearity (34) admits a global weak solution:

$$u(t,x)\in {\mathrm{L}}^{\infty }((0,\infty );{\mathrm{H}}_0^1(\Omega ))\mathrm{with}{u}_t(t,x)\in {\mathrm{L}}^{\infty }((0,\infty );{\mathrm{L}}^2(\Omega )).$$

(35)

Further, from Theorem 3.2 in [23] it follows that if $0<E\left(0\right)<d$ and $u_0\left(x\right)\in V$, then the weak solution $u(t,x)$ blows up at infinity, i.e.

$$\underset{t\to \infty }{lim\; sup}\parallel u\left(t\right)\parallel =\infty .$$

If $0<E\left(0\right)<d$, we prove in the following statement that condition (32) in Proposition 1 gives a negative sign of $I\left(u_0\right)$. Therefore, the potential well method can be applied and, according to Theorem 3.2 in [23], the weak solution blows up at infinity, i.e., we come to the same conclusion as in Proposition 1.

Proposition 3.

Suppose $u(t,x)$ is a weak solution to problem (1) and (2) with nonlinearity (34) in the maximal existence time interval $[0,T_m)$, $0<T_m\le \infty$, and $0<E\left(0\right)<d$.

(i) 

If condition (32) holds, then $I\left(u_0\right)<0$.

(ii) 

If $I\left(u_0\right)<0$ and $T_m=\infty$,

then (25) is satisfied, i.e., there exists $b\ge 0$ such that

$$E\left(0\right)<{\displaystyle \frac{k}{4}}{\parallel u\left(b\right)\parallel }^2+{\displaystyle \frac{\sqrt{k}}{2}}(u\left(b\right),u_t\left(b\right)).$$

Proof. 

(i) Assume, by contradiction, that $u_0\in W$. Then, Theorem 3.1 in [23] gives us that $u(t,x)$ is a global one and does not blow up at infinity. On the other hand, since $T_m=\infty$ and (32) holds, according to Proposition 1, we have that the solution blows up at infinity, which contradicts (35). Hence, $I\left(u_0\right)<0$.

(ii) Since $u_0\in V$, we obtain from Theorem 3.2 in [23] that $u(t,x)$ blows up at infinity and assertion (ii) follows from the necessity of Theorem 1(i). □

5.2. Arbitrary Positive Energy: $E\left(0\right)>0$

For problem (1) and (2) with nonlinearity (34) and arbitrary positive initial energy, we compare the sufficient condition (32) in Proposition 1 and the growth estimate (28) with the corresponding results of Theorem 3.1 in [25].

Firstly, we show that sufficient condition (32) in Proposition 1, that guarantees infinite time blow up of the global weak solution $u(t,x)$, is more general than condition (ii)(c) of Theorem 3.1 in [25], namely,

$$E\left(0\right)>0,I\left(u_0\right)<0,{\parallel u_0\parallel }^2>{\displaystyle \frac{4}{k}}E\left(0\right),k>1\mathrm{and}(u_0,u_1)>0.$$

(36)

Indeed, if initial data $u_0$, $u_1$ satisfy (36), then they necessarily satisfy condition (32). Furthermore, condition (32) does not require either a positive sign for the scalar product $(u_0,u_1)$, unlike (36), or the restrictions $I\left(u_0\right)<0$ and $k>1$. Thus, if the initial data satisfy (32) and $(u_0,u_1)\le 0$, then Proposition 1 in this paper can be applied, while Theorem 3.1(ii)(c) in [25] does not treat the case $(u_0,u_1)\le 0$ at all.

Secondly, we correlate the growth estimate (28) of Theorem 2 with the growth estimate (3.2) in [25]:

$${\parallel u\left(t\right)\parallel }^2\ge {\parallel u\left(t_0\right)\parallel }^{2}exp\left[{\displaystyle \frac{2(u\left(t_0\right),u_t\left(t_0\right))}{\parallel u\left(t_0\right){\parallel }^2}}(t-t_0)+{\displaystyle \frac{k}{4}}{(t-t_0)}^2\right],$$

(37)

for $t>t_0$, where $t_0$ is a large enough time such that ${\displaystyle \frac{k}{2}}{\parallel u\left(t\right)\parallel }^2>4E\left(0\right)$ for $t>t_0$.

When $\epsilon ={\displaystyle \frac{k}{2}}$, then estimates (28) and (37) from [25] coincide. It is important to point out that the main improvement of estimate (28) occurs when $\epsilon \in (0,{\displaystyle \frac{k}{2}})$. Then, the growth estimate (28) becomes faster than the growth estimate (37) from [25].

5.3. Existence of Initial Data Satisfying Sufficient Condition (32)

Below, we illustrate that the set of initial data $u_0$, $u_1$ satisfying condition (32) is not empty. We select two functions $w\left(x\right)$ and $v\left(x\right)$ such that

$$w\left(x\right)\in {\mathrm{H}}_0^1(\Omega ),v\left(x\right)\in {\mathrm{L}}^2(\Omega ),\parallel w\parallel \ne 0,\parallel v\parallel \ne 0,(w,v)=0.$$

For problem (1), (2) and (34), we construct the initial data as

$$u_0\left(x\right)=\mu w\left(x\right),u_1\left(x\right)=\eta v\left(x\right),$$

(38)

where $\mu$ and $\eta$ are positive constants. We show that if we choose the constants $\mu$ and $\eta$ to be sufficiently large, then the initial energy $E\left(0\right)$ satisfies condition (32) while being positive. From (15), it follows that

$$\begin{array}{c}E\left(0\right)={\displaystyle \frac{{\eta }^2}{2}}{\parallel v\parallel }^2+{\displaystyle \frac{{\mu }^2}{2}}{\parallel \nabla w\parallel }^2-{\displaystyle \frac{k{\mu }^2}{2}}{\int }_{\Omega }{w}^2\left(x\right)ln\left|\mu w\left(x\right)\right|dx+{\displaystyle \frac{k{\mu }^2}{4}}{\parallel w\parallel }^2\hfill \\ ={\displaystyle \frac{{\eta }^2}{2}}{\parallel v\parallel }^2+{\displaystyle \frac{{\mu }^2}{2}}{\parallel \nabla w\parallel }^2-{\displaystyle \frac{k{\mu }^{2}ln\mu }{2}}{\parallel w\parallel }^2-{\displaystyle \frac{k{\mu }^2}{2}}{\int }_{\Omega }{w}^2\left(x\right)ln\left|w\left(x\right)\right|dx+{\displaystyle \frac{k{\mu }^2}{4}}{\parallel w\parallel }^2.\hfill \end{array}$$

(39)

For nonlinearity (34) and initial data (38), condition (32) reads as follows:

$$E\left(0\right)<{\displaystyle \frac{k}{4}}{\parallel u_0\parallel }^2+{\displaystyle \frac{\sqrt{k}}{2}}(u_0,u_1)$$

(40)

or

$$\begin{array}{c}{\displaystyle \frac{{\eta }^2}{2}}{\parallel v\parallel }^2+{\displaystyle \frac{{\mu }^2}{2}}{\parallel \nabla w\parallel }^2-{\displaystyle \frac{k{\mu }^{2}ln\mu }{2}}{\parallel w\parallel }^2-{\displaystyle \frac{k{\mu }^2}{2}}{\int }_{\Omega }{w}^2\left(x\right)ln\left|w\left(x\right)\right|dx+{\displaystyle \frac{k{\mu }^2}{4}}{\parallel w\parallel }^2\\ <{\displaystyle \frac{k{\mu }^2}{4}}{\parallel w\parallel }^2.\end{array}$$

Hence, (40) is equivalent to

$${\displaystyle \frac{{\eta }^2}{2}}{\parallel v\parallel }^2+{\displaystyle \frac{{\mu }^2}{2}}h\left(\mu \right)<0,$$

(41)

where

$$\begin{array}{c}h\left(\mu \right)={\parallel \nabla w\parallel }^2-{kln\mu \parallel w\parallel }^2-k{\int }_{\Omega }{w}^2\left(x\right)ln\left|w\left(x\right)\right|dx.\end{array}$$

Since

$$\underset{\mu \to \infty }{lim}h\left(\mu \right)=-\infty ,$$

it is clear that for a sufficiently large $\mu$, condition (41) as well as (40) are satisfied. Once the constant $\mu$ has been selected, it is evident from the energy formula (39) that the constant $\eta$ can be chosen to be large enough such that $E\left(0\right)>0$.

6. Conclusions

• In this paper, we investigate the global weak solutions to the wave equation with combined nonlinearity. The nonlinearity (3)–(5) includes a logarithmic term and several power-type nonlinear terms. Moreover, the coefficients of the nonlinearity are non-negative functions of the space variables.
• A necessary and sufficient condition for blow up at infinity of the global weak solutions to problem (1) and (2) with nonlinearity (3)–(5) is proved in Theorem 1.
• Growth estimates for the blowing up at infinity solutions are derived; see Theorems 2 and 3.
• For arbitrary positive initial energy, a sufficient condition for blow up of the global solutions is obtained; see Proposition 1. This sufficient condition is more general than the previous ones; see Section 5.2.

The results presented in the paper are also applicable to the wave and Klein–Gordon equations with the more general nonlinearity (4), (8), (9). Note, that the nonlinearity (3)–(5), as well as the general nonlinearity (4), (8), (9), has not been studied to the best of our knowledge.

We consider as a critical point the lack of necessary and sufficient conditions for finite time blow up or boundedness of the local weak solutions to problem (1)–(5) in the case of arbitrary positive energy. Finding such conditions is therefore one of the main goals of our future research. As a natural extension of this research, we plan to apply the approach developed in this paper to the study of the Boussinesq and double dispersion equations of fourth and sixth order.