Existence and blow-up of weak solutions of a pseudo-parabolic equation with logarithmic nonlinearity

Abstract

In this paper, we prove the existence of weak solutions of a pseudo-parabolic equation with logarithmic nonlinearity in an interval $[0, T)$ by employing the Galerkin approximation method and compactness arguments. We show that the solutions become unbounded at a finite time $T^{\star}$ and find upper and lower bounds for this time.

1 Introduction

Showalter [39] has initiated the study of pseudo-parabolic equations, and subsequently, many authors have contributed to the various type of pseudo-parabolic equations. These equations explain the physical phenomena like unidirectional travel of long waves, aggregation of population, oozing of homogeneous fluids through cracked rocks, etc. Regarding the study of the existence and blow-up of solutions to pseudo-parabolic equations, Xu and Su [44] considered the following type of equation

$$ \textstyle\begin{cases} w_{t}(x, t) - \Delta w_{t} - \Delta w = w^{p} , & (x, t) \in \Omega \times (0, T), \\ w(x, t) = 0, & (x,t) \in \partial \Omega \times [0, T), \\ w(x, 0) =w_{0}(x) , & x \in \overline{\Omega ,} \end{cases} $$

(1.1)

where $\Omega \subset \mathbb{R}^{n}$ is a bounded domain with a smooth boundary, ∂Ω. The authors proved the global existence and unboundedness of solutions of (1.1) in finite time using a potential well method, variational methods, and comparison principle. This problem was also studied by Luo [30]; he obtained a lower bound for the blow-up time using a differential inequality technique. Moreover, for any $p>1$, he established an upper bound for the finite time of blow-up. Xu et al. [45] derived a new theorem to prove blow-up at the finite time and established an upper bound for the time using concavity method. Motivated by [44], Chen and Tian [8] studied (1.1) by considering logarithmic nonlinearity instead of $w^{p}$. In their work, the authors proved the existence of solutions by employing potential well method and derived blow-up at infinity and a condition for finite time blow-up. Han [20] derived a criterion for finite time blow-up of solutions of (1.1) by considering a general nonlinearity $f(w)$ and established an upper bound for blow-up time using the known concavity method.

Sun et al. [41] considered the pseudo-parabolic equation of the form

$$ \textstyle\begin{cases} w_{t}(x, t) - a\Delta w_{t} - \Delta w + b w = k(t) \vert w \vert ^{p-2}w , & (x, t) \in \Omega \times (0, T), \\ w(x, t) = 0, & (x,t) \in \partial \Omega \times [0, T), \\ w(x, 0) =w_{0}(x) , & x \in \overline{\Omega}, \end{cases} $$

(1.2)

where $k(t)> 0$, $a \geq 0$, $b> \lambda _{1},\lambda _{1}$ being the principal eigenvalue of −Δ. The authors analyzed the unboundedness of solutions at finite time under super-critical, critical, and sub-critical initial energy levels. Using potential wells, differential inequalities, and concavity method, they found bounds for blow-up time. The problem (1.2) has already been analyzed by Zhu et al. [46] for $a, b = 1$ and $k(t) \equiv 1$; they have established global existence and unboundedness of solutions at finite time. For the solutions of pseudo-parabolic equations with source term depending on gradient term, blow-up properties are analyzed in [31]. Sufficient conditions on the coefficients are introduced in order to specify bounded and blow-up cases, and a lower bound for blow-up time is explicitly found. Blow- up phenomena of the equation in which the source term depends only on the solution are probed, and upper and lower bounds and a blow-up criterion under specific conditions were obtained in [36].

Meyvaci [32] studied the asymptotic behavior of solutions of a pseudo-parabolic equation and in [33] generalized the study by incorporating a bounded function involving gradient term and observed the conditions under which the solution does not blow-up. Also, the author studied the finite time unboundedness of solutions and obtained lower and upper bounds for the time. Lian et al. [26] considered an initial boundary value problem of pseudo-parabolic equation with singular potential and derived global existence, asymptotic behavior, and blow-up of solutions with initial energy. Moreover, they estimated an upper bound of the blow-up time. For a nonlocal source, Wang and Xu [43] investigated a semilinear pseudo-parabolic equation for all the three initial energy levels. For subcritical and critical initial energy cases, the authors obtained results on existence, uniqueness, asymptotic behavior, and blow-up of solutions. Also, they proved that the solutions blow-up for super-critical initial energy.

Di et al. [10] studied the Dirichlet problem of the following equation

$$ \textstyle\begin{cases} w_{t} - \nu \Delta w_{t} - \operatorname{div}( \vert \nabla w \vert ^{m(x)-2}\nabla w) = \vert w \vert ^{p(x)-2}w , & (x, t) \in \Omega \times (0, T), \\ w(x, t) = 0, & (x,t) \in \partial \Omega \times [0, T), \\ w(x, 0) =w_{0}(x) , & x \in \overline{\Omega}, \end{cases} $$

(1.3)

where $p(x)$ and $m(x)$ are continuous variable exponents, and $\nu >0$. They have stated a theorem on the existence of solution to (1.3) and proved the unboundedness of solutions in finite time. They established an upper bound for the time using Kaplan’s first eigenvalue method while the lower bound is acquired by a differential inequality. Liao et al. [29] improved the results of [10] by answering some unsolved questions therein. They used Galerkin’s approximation technique to show the global existence of solutions for $p^{+} \leq 2$ and negative initial energy and presented results on nonextinction of these solutions. In [28], Liao then analyzed the case of positive initial energy and proved the nonexistence of a global solution.

All the above discussed investigations motivated us to work on the problem (1.6). The equation we consider is not only a pseudo-parabolic one but also involves logarithmic nonlinearity. Equations involving logarithmic nonlinearity are widely applied in nuclear physics, geophysics, and optics [2, 4, 16]. They appear naturally in inflation cosmology, supersymmetric field theories, and quantum mechanics [1, 13]. Looking at the very close history of problems having logarithmic nonlinearity, Chen [7] considered the following problem

$$ \textstyle\begin{cases} w_{t} - \Delta w = w \log \vert w \vert , & x \in \Omega , t > 0, \\ w(x, t) = 0, & x \in \partial \Omega , t>0, \\ w(x, 0) =w_{0}(x). & x \in \overline{\Omega}. \end{cases} $$

(1.4)

By setting a family of potential wells, he derived the global existence of solutions. In addition, decay estimates for these solutions are obtained, and the solutions for getting suitable conditions for blow-up at infinity are analyzed. Later, Han [23] improved the results of [7]. He established the criterion for the existence of global weak solutions and demonstrated the unboundedness of solutions in finite time by the concavity method. In [8], Chen and Tian studied a pseudo-parabolic problem with the same source term and boundary condition. They established the existence of global solutions, blow-up at infinity, and asymptotic behaviour of solutions under particular assumptions. In these works, the authors concluded that the presence of polynomial nonlinearity is important for the solutions to blow-up at a finite time.

However, then the studies took a turn and scientists used a more powerful logarithmic nonlinearity in their works, which made the solutions blow-up in finite time. A p-Laplacian parabolic equation with the source term $|w|^{p-2}w\log|w|$ was studied in [21, 35]. Using the potential well method and logarithmic Sobolev inequality, the authors obtained the existence and nonexistence of global solutions. They provided sufficient conditions for the finite time blow-up of solutions. Nhan and Truong [34] established existence and finite time blow-up results for the generalized equation

$$ \textstyle\begin{cases} w_{t} - \Delta w_{t} - \operatorname{div}( \vert \nabla w \vert ^{p-2}\nabla w) = w^{q-2} \log \vert w \vert , & x \in \Omega , t > 0, \\ w(x, t) = 0, & x \in \partial \Omega , t>0, \\ w(x, 0) =w_{0}(x), & x \in \overline{\Omega}, \end{cases} $$

(1.5)

when $p = q$. Cao and Liu [6] also introduced a family of potential wells to prove the global existence of solutions of (1.5) for $q = p$, and the logarithmic Sobolev inequality is used to show the blow-up of solutions at infinity. In addition, they established some decay and growth estimates and analyzed the behavior of solutions. Then, the problem (1.5) for $p < q$ was examined in [9, 12, 22]. He et al. [22] obtained finite time blow-up and decay results for weak solutions by setting a family of potential wells and using the concavity method under the condition $2 < p < q < p (1 + \frac{2}{n} )$. Ding and Zhou [9] considered more general assumptions on p and q and classified the ranges of p and q into cases under which global existence of weak solutions, finite time blow-up, and blow-up at infinity are explicitly determined for sub-critical and critical initial energy. The case of super-critical initial energy was discussed in detail by Dai et al. [12]. Lian and Xu [27] examined an initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three different initial energy levels. They proved the local existence of weak solution using contraction mapping principle and global existence, decay and infinite time blow-up using potential well method. The global well-posedness of a Kirchhoff-type wave system with logarithmic nonlinearities and weak damping was investigated by Wang et al. [42]. They obtained several results and sufficient conditions for the existence and unboundedness of solutions at different initial energy levels using potential well method and concavity method. In [25], the authors studied a semilinear wave equation with logarithmic nonlinearity and arrived at results on the existence and blow-up of solutions using potential well method. Weak solutions and blow-up of different partial differential equations are discussed in [3, 17–19, 38].

Based on the above-mentioned works and motivated by [10, 22], we are excited to study the existence and blow-up of weak solutions of the following pseudo-parabolic equation with logarithmic nonlinearity

$$ \textstyle\begin{cases} w_{t} - \Delta w_{t} - \operatorname{div}( \vert \nabla w \vert ^{p(x)-2}\nabla w) \\ \quad = \vert w \vert ^{s(x)-2}w + \vert w \vert ^{h-2}w\log \vert w \vert , & (x, t) \in \Omega \times (0, \infty ), \\ w(x, t) = 0, & (x,t) \in \partial \Omega \times [0, \infty ), \\ w(x, 0) =w_{0}(x), & x \in \overline{\Omega}, \end{cases} $$

(1.6)

where $\Omega \subset \mathbb{R}^{n}(n \geq 1)$ is a bounded domain with smooth boundary ∂Ω. The model considered in (1.6) is used to describe the non-stationary process in semiconductors in the presence of sources; the first two terms represent the free-electron density rate and logarithmic and polynomial nonlinearity stands for the source of free-electron current [24]. The motivation of this work is to address the existence and finite time blow-up of solutions of the non-stationary process in semiconductors in the presence of logarithmic and polynomial sources.

The log-Hölder continuous variable exponents $p(x)$, $s(x)$ and the constant h satisfy the following hypotheses

• $$ 2 \leq p_{-} \leq p(x) \leq p_{+}< s_{-}\leq s(x)\leq s_{+} < h < \infty , $$

  (1.7)
• $$ p^{*}(x) > 2, \qquad p^{*}(x) = \textstyle\begin{cases} \frac{n p(x)}{n - p(x)}, & p(x)< n, \\ \infty , & p(x) \geq n, \end{cases} $$

  (1.8)
• $$ \operatorname*{ess\,inf}_{x\in \Omega} \bigl(p^{*}(x) - s(x) \bigr) > 0. $$

  (1.9)

This paper is arranged as follows: In Sect. 2, we state the required preliminaries. The existence results are discussed in Sect. 3 using the Faedo-Galerkin approximation method. Blow-up analysis of the solutions is done in Sects. 4 and 5. In this paper, C and $C(\epsilon )$ are generic constants, which may vary accordingly.

2 Preliminaries

To discuss the problem (1.6), we need the following facts about generalized Lebesgue and Sobolev spaces. For more details, one can refer to [11]. In this section, we take $p, s : \Omega \longrightarrow [1, \infty )$ as measurable functions, and $\Omega \subset \mathbb{R}^{n}$ is bounded.

Definition 2.1

([5])

Let X be a Banach space. Then $L^{p}(0, T, X)$ is defined as the set of measurable functions $w : [0, T]\longrightarrow X$ such that

if $1\leq p <\infty$,

$$ \Vert w \Vert _{L^{p}(0, T; X)} = \biggl( \int _{0}^{T} \bigl\Vert w(t) \bigr\Vert _{X}^{p} \,dt \biggr)^{\frac{1}{p}} < \infty , $$

and if $p = \infty$,

$$ \Vert w \Vert _{L^{\infty}(0, T; X)} = \operatorname*{ess\,sup}_{0\leq t\leq T} \bigl\Vert w(t) \bigr\Vert _{X} < \infty . $$

Remark 2.1

For $1 \leq p \leq \infty$, $L^{p}(0, T; X)$ is a Banach space with the above norms.

Definition 2.2

([11])

The variable exponent Lebesgue space with exponent $p(x)$ is defined by

$$ L^{p(x)}(\Omega ) := \bigl\{ w:\Omega \longrightarrow \mathbb{R}| \rho _{p(x)}( \lambda w)< \infty , \text{ for some } \lambda >0\bigr\} , $$

where

$$ \rho _{p(x)}(w) = \int _{\Omega}\bigl\vert w(x) \bigr\vert ^{p(x)} \,dx. $$

Theorem 2.1

([11])

The space $L^{p(x)}(\Omega )$ endowed with the Luxembourg norm

$$ \Vert w \Vert _{p(x)} = \inf \biggl\{ \lambda >0 | \rho _{p(x)} \biggl( \frac{w}{\lambda} \biggr)\leq 1\biggr\} , $$

is a Banach space and

$$ \min \bigl\{ \Vert w \Vert _{p(x)}^{p_{-}}, \Vert w \Vert _{p(x)}^{p_{+}}\bigr\} \leq \int _{\Omega} \vert w \vert ^{p(x)} \,dx \leq \max \bigl\{ \Vert w \Vert _{p(x)}^{p_{-}}, \Vert w \Vert _{p(x)}^{p_{+}} \bigr\} $$

where $p_{-} = \min p(x)$ and $p_{+} =\max p(x)$ on Ω.

Remark 2.2

([11])

$L^{p'(x)}(\Omega )$ denotes the dual space of $L^{p(x)}(\Omega )$ with $\frac{1}{p(x)}+\frac{1}{p'(x)}=1$.

Definition 2.3

([11])

The variable exponent Sobolev space is defined as

$$ W^{k, p(x)}(\Omega ) = \bigl\{ w\in L^{p(x)}(\Omega )|D^{\alpha}w\in L^{p(x)}( \Omega ), \vert \alpha \vert \leq k \bigr\} , $$

where $k\geq 1$, $D^{\alpha}w$ is the $\alpha ^{th}$ weak partial derivative with $\alpha = (\alpha _{1}, \alpha _{2}, \ldots,\alpha _{N})$, a multi-index and $|\alpha |= \sum_{j=1}^{N} \alpha _{j}$.

Theorem 2.2

([11])

The variable exponent Sobolev space $W^{k, p(x)}(\Omega )$ endowed with the norm $\|w\|_{k, p(x)}:= \sum_{|\alpha |\leq k}\|D^{\alpha}w\|_{p(x)}$ is a Banach space.

Observe that $W^{k, p(x)}_{0}(\Omega )$ is the closure of $C_{0}^{\infty}(\Omega )$ in $W^{k, p(x)}(\Omega )$.

Lemma 2.1

([11])

If $p(x)$, $s(x)$ are variable exponents satisfying $p(x) \leq s(x)$ a.e. in Ω, then there is a continuous embedding from $L^{s(x)}(\Omega )\hookrightarrow L^{p(x)}(\Omega )$.

Lemma 2.2

There exists a continuous and compact Sobolev embedding $W_{0}^{1, p(x)}(\Omega ) \hookrightarrow L^{s(x)}(\Omega )$, where the variable exponents $p(x)\in C(\overline{\Omega})$, $s: \Omega \longrightarrow [1, \infty )$ are measurable functions and satisfy

$$ \operatorname*{ess\,inf}_{x\in \overline{\Omega}}\bigl(p^{*}(x) - s(x)\bigr)> 0, \quad \textit{where } p^{*} = \textstyle\begin{cases} \frac{n p(x)}{n-p(x)}, & \textit{if } p(x) < n, \\ \infty , & \textit{if } p(x) \geq n. \end{cases} $$

Lemma 2.3

([9, 37] [lemma 2])

For all $w \in [1, \infty )$

$$ \vert \log w \vert \leq \frac{w^{\eta}}{e\eta}, $$

where η is a positive number.

3 Weak solutions

Here we prove the existence of weak solutions to the equation (1.6). The main result Theorem 3.1 can be proved using the Faedo-Galerkin approximation method and Sobolev embeddings as in [22].

Definition 3.1

A function $w \in L^{2}(0,T; W^{1, p(x)}_{0}(\Omega )\cap L^{s(x)}(\Omega )) \cap L^{\infty}(0,T; H^{1}_{0}(\Omega ) )\cap C(0,T; H^{1}_{0}( \Omega ))$ is said to be a weak solution to (1.6) if $w_{0} \in W^{1, p(x)}_{0}(\Omega )\backslash \{0\}$, $w_{t} \in L^{2}(0,T; H^{1}_{0}(\Omega ))$, and w satisfies

$$\begin{aligned}& \int _{0}^{T} \int _{\Omega}w_{t} \phi\, dx\,dt + \int _{0}^{T} \int _{\Omega}\nabla w_{t} \nabla \phi \,dx \,dt + \int _{0}^{T} \int _{\Omega} \vert \nabla w \vert ^{p(x)-2}\nabla w \nabla \phi \,dx \,dt \\& \quad = \int _{0}^{T} \int _{\Omega} \vert w \vert ^{s(x)-2}w \phi \,dx \,dt + \int _{0}^{T} \int _{\Omega} \vert w \vert ^{h-2}w \log \vert w \vert \phi \,dx \,dt, \end{aligned}$$

(3.1)

$\forall \phi \in C^{\infty}(0,T; C_{0}^{\infty}(\Omega ))$.

Definition 3.2

For $w_{0} \in W^{1, p(x)}_{0}(\Omega )\backslash \{0\}$, define an energy functional as

$$\begin{aligned} \mathfrak{N}(w) = - \int _{\Omega}\frac{ \vert w \vert ^{s(x)}}{s(x)} \,dx + \int _{\Omega}\frac{ \vert \nabla w \vert ^{p(x)}}{p(x)} \,dx + \frac{1}{h^{2}} \int _{\Omega} \vert w \vert ^{h} \,dx - \frac{1}{h} \int _{\Omega} \vert w \vert ^{h} \log \vert w \vert \,dx. \end{aligned}$$

(3.2)

Theorem 3.1

Suppose $w_{0} \in W^{1, p(x)}_{0}(\Omega )\backslash \{0\}$ and $p(x)$, $s(x)$, h satisfy the conditions (1.7), (1.8), and (1.9). Then, the equation (1.6) has a weak solution.

Proof

Now we consider an orthonormal basis of $L^{2}(\Omega )$, which is orthogonal in $H_{0}^{1}(\Omega )$ given by $\{r_{j}\}_{j=1}^{\infty}$ and a collection of eigenfunctions of Δ corresponding to the eigenvalues $\{\lambda _{j}\}_{j=1}^{\infty}$. We seek for finite dimensional approximation solutions to (1.6) as

$$ w_{m} = \sum_{j=1}^{m} c_{mj}(t)r_{j}(x), $$

(3.3)

where $c_{mj}$ are unknown and satisfy

$$\begin{aligned}& \int _{\Omega}w'_{m} r_{i} \,dx + \int _{\Omega}\nabla w'_{m} \nabla r_{i} \,dx + \int _{\Omega} \vert \nabla w_{m} \vert ^{p(x)-2}\nabla w_{m}\nabla r_{i} \,dx \\& \quad = \int _{\Omega} \vert w_{m} \vert ^{s(x)-2}w_{m} r_{i} \,dx + \int _{\Omega} \vert w_{m} \vert ^{h-2}w_{m} \log \vert w_{m} \vert r_{i} \,dx, \end{aligned}$$

(3.4)

and

$$ w_{0m} = \sum_{j=1}^{m} c_{mj}(0)r_{j}(x) \longrightarrow w_{0} \quad \text{in } W^{1, p(x)}_{0}(\Omega ). $$

(3.5)

This generates an initial value problem for a system of ordinary differential equations in $\{c_{mi}(t)\}_{i=1}^{m}$, namely,

$$ \textstyle\begin{cases} (1+\lambda _{i})\frac{d}{dt} c_{mi}(t)= F(c_{m1}, c_{m2}, \ldots, c_{mm}), \\ c_{mi}(0)= (w_{0}, r_{i})_{L^{2}}, \end{cases} $$

where

$$\begin{aligned}& F(c_{m1}, c_{m2}, \ldots, c_{mm})\\& \quad = \int _{\Omega}\bigl( - \vert \nabla w_{m} \vert ^{p(x)-2} \nabla w_{m} \nabla r_{i} + \vert w_{m} \vert ^{s(x)-2} w_{m} r_{i} + \vert w_{m} \vert ^{h-2}w_{m} \log \vert w_{m} \vert r_{i} \bigr)\,dx . \end{aligned}$$

Since, $F(c_{m1}, c_{m2}, \ldots, c_{mm})$ depends on $(c_{m1}, c_{m2}, \ldots, c_{mm})$ continuously, Peano’s theorem gives the existence of a local solution to this problem.

Now multiply (3.4) by $c_{mi}(t)$ and sum over i to get

$$\begin{aligned}& \int _{\Omega}w'_{m} w_{m} \,dx + \int _{\Omega}\nabla w'_{m} \nabla w_{m} \,dx + \int _{\Omega} \vert \nabla w_{m} \vert ^{p(x)-2}\nabla w_{m} \nabla w_{m} \,dx \\& \quad = \int _{\Omega} \vert w_{m} \vert ^{s(x)-2} w_{m} w_{m} \,dx + \int _{\Omega} \vert w_{m} \vert ^{h-2}w_{m} \log \vert w_{m} \vert w_{m} \,dx. \end{aligned}$$

(3.6)

This gives

$$ \begin{gathered}[b] \frac{d}{dt} \biggl( \frac{1}{2} \int _{\Omega} \vert w_{m} \vert ^{2} + \vert \nabla w_{m} \vert ^{2} \,dx + \int _{0}^{t} \int _{\Omega}\bigl\vert \nabla w_{m}(x, \tau ) \bigr\vert ^{p(x)}\,dx \,d \tau \biggr) \\ \quad = \int _{\Omega} \vert w_{m} \vert ^{s(x)} \,dx + \int _{\Omega} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx. \end{gathered} $$

(3.7)

Integrating (3.7) over $(0, t)$, we get

$$ \mathcal{H}_{m}(t) = \mathcal{H}_{m}(0) + \int _{0}^{t} \int _{\Omega} \vert w_{m} \vert ^{s(x)} \,dx \,d\tau + \int _{0}^{t} \int _{\Omega} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx \,d\tau , $$

(3.8)

where $\mathcal{H}_{m}(t)= \frac{1}{2} \int _{\Omega}|w_{m}|^{2} + |\nabla w_{m}|^{2} \,dx + \int _{0}^{t} \int _{\Omega}|\nabla w_{m}|^{p(x)} \,dx \,d\tau$.

Now, we look for estimates to (3.8). By Lemma 2.3 and following the calculations similar to [22], we get

$$\begin{aligned}& \int _{\Omega} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx \\ & \quad \leq \int _{\{x\in \Omega : \vert w_{m} \vert < 1 \}} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx + \int _{\{x \in \Omega : \vert w_{m} \vert \geq 1 \}} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx \\ & \quad \leq \int _{\{x\in \Omega : \vert w_{m} \vert \geq 1 \}} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx, \quad \bigl[\text{since } \log \vert w_{m} \vert < 0 \text{ for } \vert w_{m} \vert < 1\bigr] \\ & \quad \leq \frac{1}{e\eta} \int _{\{x\in \Omega : \vert w_{m} \vert \geq 1 \}} \vert w_{m} \vert ^{h+ \eta} \,dx \\ & \quad \leq \frac{1}{e\eta} \Vert w_{m} \Vert _{h+\eta}^{h+\eta}. \end{aligned}$$

(3.9)

Choose η such that $p_{-}< \eta < \frac{np_{-}}{n - p_{-}} - h$. Then, by the interpolation inequality, we obtain

$$\begin{aligned} \int _{\Omega} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx \leq & C \Vert w_{m} \Vert _{2}^{(1- \theta )(h+\eta )} \Vert w_{m} \Vert _{\frac{np_{-}}{n - p_{-}}}^{\theta (h+ \eta )}, \end{aligned}$$

where $\theta \in (0, 1)$ is given by $\frac{1}{h+ \eta} = \frac{\theta (n - p_{-})}{np_{-}} + \frac{1-\theta}{2}$. We have the following continuous embeddings $W^{1, p(x)}_{0}(\Omega ) \hookrightarrow L^{p^{*}(x)}(\Omega )$ and $L^{p^{*}(x)}(\Omega ) \hookrightarrow L^{\frac{np_{-}}{n - p_{-}}}( \Omega )$ that together give

$$ \int _{\Omega} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx \leq C \Vert w_{m} \Vert _{2}^{(1- \theta )(h+\eta )} \Vert w_{m} \Vert _{W^{1, p(x)}_{0}(\Omega )}^{\theta (h + \eta )}. $$

Here we assume $\|w_{m}\|_{W^{1, p(x)}_{0}(\Omega )} \geq 1$. Then by Theorem 1.3 of [14] and following the calculations similar to [15], we get

$$ \int _{\Omega} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx \leq C \Vert w_{m} \Vert _{2}^{(1- \theta )(h+\eta )} \biggl( \int _{\Omega} \vert \nabla w_{m} \vert ^{p(x)} \,dx \biggr)^{\frac{\theta (h+\eta )}{p_{-}}}. $$

Since $p_{-} \geq 2$, $\frac{\theta (h+\eta )}{p_{-}} < 1$. Similarly, if $\|w_{m}\|_{W^{1, p(x)}_{0}(\Omega )} < 1$, we get

$$ \int _{\Omega} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx \leq C \Vert w_{m} \Vert _{2}^{(1- \theta )(h+\eta )} \biggl( \int _{\Omega} \vert \nabla w_{m} \vert ^{p(x)} \,dx \biggr)^{\frac{\theta (h+\eta )}{p_{+}}}. $$

However, in this paper, we proceed with the calculations under the assumption $\|w_{m}\|_{W^{1, p(x)}_{0}(\Omega )} \geq 1$, since we can do the other case in the same way. Now employing Young’s inequality with $\epsilon > 0$, we obtain

$$ \int _{\Omega} \vert w_{m} \vert ^{h} \log \vert w_{m} \vert \,dx \leq C( \epsilon ) \bigl( \Vert w_{m} \Vert _{2}^{2} \bigr)^{\nu}+ \epsilon \int _{\Omega} \vert \nabla w_{m} \vert ^{p(x)} \,dx, $$

(3.10)

where $\nu = \frac{(1-\theta )(h+\eta )p_{-}}{2p_{-} - 2\theta (h+ \eta )} > 1$.

To proceed further choose the sets $\Omega _{1}^{+} = \{x\in \Omega : |w_{m}|\geq 1\}$ and $\Omega _{1}^{-} = \{x\in \Omega :| w_{m}|< 1 \}$. Hence, we can write

$$ \int _{\Omega} \vert w_{m} \vert ^{s(x)} \,dx \leq \int _{\Omega _{1}^{-}} \vert w_{m} \vert ^{s_{-}} \,dx + \int _{\Omega _{1}^{+}} \vert w_{m} \vert ^{s_{+}} \,dx \leq \Vert w_{m} \Vert _{s_{-}}^{s_{-}}+ \Vert w_{m} \Vert _{s_{+}}^{s_{+}}. $$

(3.11)

This gives

$$ \int _{\Omega} \vert w_{m} \vert ^{s(x)} \,dx \leq 2 \Vert w_{m} \Vert _{s_{+}}^{s_{+}}. $$

(3.12)

Applying the Gagliardo-Nirenberg interpolation inequality to (3.12), we get

$$ \Vert w_{m} \Vert _{s_{+}}^{s_{+}} \leq C \Vert \nabla w_{m} \Vert _{p_{-}}^{\vartheta s_{+}} \Vert w_{m} \Vert _{2}^{(1-\vartheta )s_{+}}, $$

(3.13)

where $\vartheta = \frac{(2-s_{+})np_{-}}{s_{+}(2n - 2p_{-} -np_{-})}$. Now Young’s inequality gives us

$$ \Vert w_{m} \Vert _{s_{+}}^{s_{+}} \leq \epsilon \Vert \nabla w_{m} \Vert _{p_{-}}^{p_{-}} + C(\epsilon ) \Vert w_{m} \Vert _{2}^{ \frac{p_{-}(1-\vartheta )s_{+}}{p_{-} -\vartheta s_{+}}} , \quad \epsilon >0. $$

(3.14)

Application of the inequalities (3.10) and (3.11) together with (3.14) in (3.8) gives

$$\begin{aligned} \mathcal{H}_{m}(t) \leq & \mathcal{H}_{m}(0) + \epsilon \int _{0}^{t} \Vert \nabla w_{m} \Vert _{p_{-}}^{p_{-}} \,d\tau + C(\epsilon ) \int _{0}^{t} \Vert w_{m} \Vert _{2}^{ \frac{p_{-}(1-\vartheta )s_{+}}{p_{-} - \vartheta s_{+}}} \,d\tau \\ & + C(\epsilon ) \int _{0}^{t} \bigl( \Vert w_{m} \Vert _{2}^{2} \bigr)^{\nu}\,d\tau + \epsilon \int _{0}^{t} \int _{\Omega} \vert \nabla w_{m} \vert ^{p(x)} \,dx \,d\tau . \end{aligned}$$

Let $\delta = \text{ max }\{2\nu , { \frac{p_{-}(1-\vartheta )s_{+}}{p_{-} -\vartheta s_{+}}} \}$. Now, by Lemma 2.1 and Theorem 2.1, we get

$$ \mathcal{H}_{m}(t) \leq \mathcal{H}_{m}(0) + 2\epsilon \int _{0}^{t} \int _{\Omega} \vert \nabla w_{m} \vert ^{p(x)} \,dx \,d\tau + C(\epsilon ) \int _{0}^{t} \Vert w_{m} \Vert _{2}^{\delta}\,d\tau . $$

(3.15)

Further, by putting $\epsilon = \frac{1}{4}$ and using the definition of $\mathcal{H}_{m}(t)$, we arrive at the following inequality

$$ \mathcal{H}_{m}(t) \leq 2\mathcal{H}_{m}(0) + C \int _{0}^{t} \mathcal{H}_{m}^{\delta}(s) \,d\tau . $$

To carry forward, we apply the Gronwall-Bellman-Bihari-type integral inequality and obtain

$$ \mathcal{H}_{m}(t) \leq C_{T}, $$

(3.16)

where the constant $C_{T}$ depends on T. Hence

$$ \frac{1}{2} \int _{\Omega}\bigl( \vert w_{m} \vert ^{2}+ \vert \nabla w_{m} \vert ^{2} \bigr) \,dx + \int _{0}^{T} \int _{\Omega} \vert \nabla w_{m} \vert ^{p(x)}\,dx \,d\tau \leq C_{T}. $$

(3.17)

Assuming $\operatorname{min} \{\|\nabla w_{m}\|_{p(x)}^{p_{-}},\| \nabla w_{m}\|_{p(x)}^{p_{+}} \} = \|\nabla w_{m}\|_{p(x)}^{p_{-}}$, by (3.17), (1.7) and Theorem 2.1, we get

$$ \int _{0}^{T} \Vert \nabla w_{m} \Vert _{p(x)}^{2} \,d\tau \leq \int _{0}^{T} \int _{\Omega} \vert \nabla w_{m} \vert ^{p(x)}\,dx \,d\tau \leq C_{T}. $$

(3.18)

Now consider the functional $\mathfrak{N}(w)$ defined in definition (3.2). Since it is continuous and we have $w_{0m} \longrightarrow w_{0}$ in $W_{0}^{1, p(x)}(\Omega )$, we get a constant C with

$$ \mathfrak{N}(w_{0m}) \leq C, $$

(3.19)

for any integer $m > 0$ large enough.

Multiplying (3.4) by $c'_{mi}(t)$ and summing over i, then integrating with respect to t gives

$$ \int _{0}^{t} \bigl\Vert w'_{m}(s) \bigr\Vert _{H_{0}^{1}(\Omega )}^{2} \,d\tau + \mathfrak{N} \bigl(w_{m}(t)\bigr) = \mathfrak{N}\bigl(w_{m}(0)\bigr). $$

The inequality (3.19) gives

$$ \int _{0}^{t} \bigl\Vert w'_{m}(s) \bigr\Vert _{H_{0}^{1}(\Omega )}^{2} \,d\tau + \mathfrak{N} \bigl(w_{m}(t)\bigr) \leq C. $$

(3.20)

From the estimates (3.17), (3.18), and (3.20), together with the standard compactness arguments, we get

$$\begin{aligned}& w_{m} \longrightarrow w\quad \text{$\mathrm{weakly} ^{*}$ in } L^{\infty}\bigl(0,T; H_{0}^{1}(\Omega ) \bigr), \end{aligned}$$

(3.21)

$$\begin{aligned}& w_{m} \longrightarrow w \quad \text{weakly in } L^{2} \bigl(0,T; W^{1,p(x)}_{0}( \Omega ) \bigr), \end{aligned}$$

(3.22)

$$\begin{aligned}& w'_{m} \longrightarrow w' \quad \text{weakly in } L^{2} \bigl(0, T; H_{0}^{1}( \Omega ) \bigr), \end{aligned}$$

(3.23)

$$\begin{aligned}& \vert \nabla w_{m} \vert ^{p(x)-2}\nabla w_{m} \longrightarrow \xi \quad \text{weakly in } L^{2} \bigl(0, T; L^{ p'(x)}(\Omega ) \bigr). \end{aligned}$$

(3.24)

Since $w_{m} \in W^{1, p(x)}_{0}(\Omega )$, the Sobolev embedding gives

$$ \int _{0}^{T} \Vert w_{m} \Vert _{s(x)}^{2} \,d\tau \leq C \int _{0}^{T} \Vert \nabla w_{m} \Vert _{p(x)}^{2} \,d\tau \leq C_{T} \quad \text{ by (3.22)}. $$

This implies

$$ w_{m} \rightharpoonup w \quad \text{in } L^{2} \bigl(0, T; L^{s(x)}( \Omega ) \bigr). $$

Since we have the convergences (3.21) and (3.23), by employing the Aubin-Lions lemma [40], we get

$$ w_{m} \longrightarrow w \quad \text{in } C \bigl(0, T; L^{2}(\Omega ) \bigr), $$

which implies

$$ w_{m} \longrightarrow w \quad \text{a.e. on }\Omega \times (0, T). $$

Thus, we get

$$\begin{aligned} \begin{gathered} \vert w_{m} \vert ^{h-2}w_{m} \log \vert w_{m} \vert \longrightarrow \vert w \vert ^{h-2}w \log \vert w \vert \quad \text{a.e. on }\Omega \times (0, T), \\ \vert w_{m} \vert ^{s(x)-2}w_{m} \longrightarrow \vert w \vert ^{s(x)-2}w \quad \text{a.e. on } \Omega \times (0, T). \end{gathered} \end{aligned}$$

(3.25)

Since we have $p_{-}< \eta < \frac{np_{-}}{n - p_{-}} - h$, we can choose $\gamma >0$ such that $p_{-}<(h-1+\gamma )h' < p^{*}$. Now, following the trick in [22], we get

$$\begin{aligned} \int _{\Omega}\bigl\vert \psi _{m}(x,t) \bigr\vert ^{h'} \,dx =& \int _{\Omega ^{-}_{1}} \bigl\vert \psi _{m}(x,t) \bigr\vert ^{h'} \,dx + \int _{\Omega ^{+}_{1}} \bigl\vert \psi _{m}(x,t) \bigr\vert ^{h'} \,dx \\ \leq & \bigl(e(h-1)\bigr)^{-h'} \vert \Omega \vert + ( \gamma )^{-h'} \int _{\Omega _{1}^{+}} \bigl( \vert w_{m} \vert ^{h-1+\gamma}\bigr)^{h'}\,dx \\ = & \bigl(e(h-1)\bigr)^{-h'} \vert \Omega \vert + (\gamma )^{-h'} \Vert w_{m} \Vert _{(h-1+\gamma )h'}^{(h-1+ \gamma )h'}, \end{aligned}$$

(3.26)

where $\psi _{m}(x,t)= |w_{m}(x,t)|^{h-1}\log |w_{m}(x,t)|$. Choosing $\eta = \frac{\gamma h}{h-1}$ in (3.9) and following the calculations up to (3.10), we get

$$ \int _{\Omega}\bigl\vert \psi _{m}(x,t) \bigr\vert ^{h'} \,dx \leq C(\epsilon ) \bigl( \Vert w_{m} \Vert _{2}^{2} \bigr)^{\nu}+ \epsilon \int _{\Omega} \vert \nabla w_{m} \vert ^{p(x)} \,dx. $$

(3.27)

Integrating this inequality over $(0,T)$ and applying (3.17), we get

$$ \int _{0}^{T} \int _{\Omega}\bigl\vert \psi _{m}(x,t) \bigr\vert ^{h'} \,dx\,dt \leq C_{T}. $$

(3.28)

Also,

$$ \int _{0}^{T} \int _{\Omega}\bigl( \vert w_{m} \vert ^{s(x)-1}\bigr)^{s'(x)}\,dx \leq \int _{0}^{T} \int _{\Omega} \vert w_{m} \vert ^{s(x)}\,dx \leq C_{T}. $$

(3.29)

Hence, from (3.25), (3.29), and Lion’s lemma (see [40], Lemma 1.3, p.12), we have

$$ \vert w_{m} \vert ^{h-2}w_{m} \log \vert w_{m} \vert \longrightarrow \vert w \vert ^{h-2}w \log \vert w \vert \quad \text{weakly star in } L^{\infty}\bigl(0,T;L^{h^{\prime}}(\Omega ) \bigr)$$

Now, since we have the monotonicity of $|\zeta |^{p(x)-2}\zeta$, making use of the Minty-Browder condition, we get $\xi =|\nabla w|^{p(x)-2}\nabla w$. Hence the proof. □

4 Upper bound for blow-up time

Here our objective is to seek an upper bound for the time at which the solutions to the problem (1.6) become unbounded.

Theorem 4.1

Let w be a weak solution of (1.6) and assume that $w_{0}$ satisfies

$$ \int _{\Omega}\biggl(\frac{ \vert w_{0} \vert ^{s(x)}}{s(x)}- \frac{ \vert \nabla w_{0} \vert ^{p(x)}}{p(x)} \biggr)\,dx -\frac{1}{h^{2}} \int _{\Omega} \vert w_{0} \vert ^{h} \,dx + \frac{1}{h} \int _{\Omega} \vert w_{0} \vert ^{h} \log \vert w_{0} \vert \,dx \geq 0. $$

(4.1)

Then the solution w blows up at a finite time $T^{\star}> 0$. In addition, there exists an upper bound for the time as given below

$$ T^{\star}\leq \frac{2 [N(0) ]^{ (\frac{2-p_{-}}{2} )}}{(p_{-} -2)\theta}, $$

(4.2)

where $\theta > 0$ is some constant.

Proof

We have the energy functional related to the problem (1.6) given by

$$\begin{aligned} \mathfrak{N}(t) = - \int _{\Omega}\frac{ \vert w \vert ^{s(x)}}{s(x)} \,dx + \int _{\Omega}\frac{ \vert \nabla w \vert ^{p(x)}}{p(x)} \,dx + \frac{1}{h^{2}} \int _{\Omega} \vert w \vert ^{h} \,dx - \frac{1}{h} \int _{\Omega} \vert w \vert ^{h} \log \vert w \vert \,dx, \end{aligned}$$

(4.3)

which gives

$$ \mathfrak{N'}(t) = - \int _{\Omega} \vert w_{t} \vert ^{2} + \vert \nabla w_{t} \vert ^{2} \,dx \leq 0. $$

(4.4)

Now we set an auxiliary functional

$$ N(t) = \int _{\Omega} \vert w \vert ^{2} + \vert \nabla w \vert ^{2} \,dx. $$

(4.5)

Multiply (1.6) by w and integrate over Ω to get

$$ \int _{\Omega}w w_{t} \,dx + \int _{\Omega}\nabla w \nabla w_{t} \,dx = \int _{\Omega} \vert w \vert ^{s(x)} \,dx - \int _{\Omega} \vert \nabla w \vert ^{p(x)} \,dx + \int _{\Omega} \vert w \vert ^{h} \log \vert w \vert \,dx. $$

(4.6)

Now differentiate $N(t)$ with respect to t to obtain

$$\begin{aligned} N'(t) =& 2 \int _{\Omega}( w w_{t} + \nabla w \nabla w_{t} )\,dx \\ =& 2 \biggl[ \int _{\Omega} \vert w \vert ^{s(x)} \,dx - \int _{\Omega} \vert \nabla w \vert ^{p(x)} \,dx + \int _{\Omega} \vert w \vert ^{h} \log \vert w \vert \,dx \biggr] \\ =& 2 \biggl[ \int _{\Omega}s(x) \biggl[\frac{ \vert w \vert ^{s(x)}}{s(x)} - \frac{ \vert \nabla w \vert ^{p(x)}}{p(x)} \biggr]\,dx - \int _{\Omega}\frac{s(x)}{h^{2}} \vert w \vert ^{h} \,dx + \int _{\Omega}\frac{s(x)}{h} \vert w \vert ^{h} \log \vert w \vert \,dx \\ & {} + \int _{\Omega}s(x) \biggl[ \frac{1}{p(x)} - \frac{1}{s(x)} \biggr] \vert \nabla w \vert ^{p(x)} \,dx + \int _{\Omega}\frac{s(x)}{h^{2}} \vert w \vert ^{h} \,dx \\ & {} - \int _{\Omega}\frac{s(x)}{h} \vert w \vert ^{h} \log \vert w \vert \,dx + \int _{\Omega} \vert w \vert ^{h} \log \vert w \vert \,dx \biggr]. \end{aligned}$$

(4.7)

Since we have $\mathfrak{N'}(t) \leq 0$, we get

$$\begin{aligned}& \int _{\Omega}s(x) \biggl[ \frac{ \vert w \vert ^{s(x)}}{s(x)} - \frac{ \vert \nabla w \vert ^{p(x)}}{p(x)} -\frac{1}{h^{2}} \vert w \vert ^{h} + \frac{1}{h} \vert w \vert ^{h} \log \vert w \vert \biggr] \,dx \\& \quad \geq s_{-} \int _{\Omega}\biggl[ \frac{ \vert w_{0} \vert ^{s(x)}}{s(x)} - \frac{ \vert \nabla w_{0} \vert ^{p(x)}}{p(x)} - \frac{1}{h^{2}} \vert w_{0} \vert ^{h} + \frac{1}{h} \vert w_{0} \vert ^{h} \log \vert w_{0} \vert \biggr] \,dx \geq 0. \end{aligned}$$

Hence

$$\begin{aligned} N'(t) \geq & 2 \biggl[ \int _{\Omega}s(x) \biggl[ \frac{1}{p(x)} - \frac{1}{s(x)} \biggr] \vert \nabla w \vert ^{p(x)} \,dx \\ &{}+ \int _{\Omega}\frac{s(x)}{h^{2}} \vert w \vert ^{h} \,dx + \int _{\Omega}\biggl(1 - \frac{s(x)}{h} \biggr) \vert w \vert ^{h} \log \vert w \vert \,dx \biggr], \end{aligned}$$

by (1.7), we know $(1 - \frac{s(x)}{h} ) >0$. So,

$$\begin{aligned} N'(t) \geq & 2 \int _{\Omega}s(x) \biggl[ \frac{1}{p(x)} - \frac{1}{s(x)} \biggr] \vert \nabla w \vert ^{p(x)} \,dx \\ \geq & 2 \int _{\Omega}s_{-} \biggl[ \frac{1}{p_{+}} - \frac{1}{s_{-}} \biggr] \vert \nabla w \vert ^{p(x)} \,dx = \beta _{1} \int _{\Omega} \vert \nabla w \vert ^{p(x)}\,dx \geq 0, \end{aligned}$$

(4.8)

where $\beta _{1} = 2 s_{-} [ \frac{1}{p_{+}} - \frac{1}{s_{-}} ]$. Now define the sets $\Omega _{2}^{+} = \{x\in \Omega : |\nabla w|\geq 1\}$ and $\Omega _{2}^{-} = \{x\in \Omega :|\nabla w|\leq 1 \}$. Since we have $\|\nabla w \|_{2} \leq C \|\nabla w\|_{\gamma}$ for all $\gamma \geq 2$, we get

$$\begin{aligned} N'(t) \geq& \beta _{1} \biggl[ \int _{\Omega _{2}^{-}} \vert \nabla w \vert ^{p_{+}} \,dx + \int _{\Omega _{2}^{+}} \vert \nabla w \vert ^{p_{-}} \,dx \biggr]\\ \geq &\beta _{2} \biggl[ \biggl( \int _{\Omega _{2}^{-}} \vert \nabla w \vert ^{2} \,dx \biggr)^{ \frac{{p_{+}}}{2}} + \biggl( \int _{\Omega _{2}^{+}} \vert \nabla w \vert ^{2} \,dx \biggr)^{\frac{{p_{-}} }{2}} \biggr]. \end{aligned}$$

This will give

$$\begin{aligned} \bigl(N'(t)\bigr)^{\frac{2}{p_{+}}} \geq & \beta _{3} \int _{\Omega _{2}^{-}} \vert \nabla w \vert ^{2} \,dx \geq 0, \end{aligned}$$

(4.9)

$$\begin{aligned} \bigl(N'(t)\bigr)^{\frac{2}{p_{-}}} \geq & \beta _{4} \int _{\Omega _{2}^{+}} \vert \nabla w \vert ^{2} \,dx \geq 0. \end{aligned}$$

(4.10)

From the Poincare inequality, we can deduce that $\|\nabla w\|_{2}^{2} \geq \kappa \|w\|_{2}^{2}$, where κ is the first eigenvalue of −Δ. Therefore, we get

$$\begin{aligned} \Vert \nabla w \Vert _{2}^{2} =& \frac{1}{1+\kappa} \Vert \nabla w \Vert _{2}^{2} + \frac{\kappa}{1+\kappa} \Vert \nabla w \Vert _{2}^{2} \\ \geq& \frac{\kappa}{1+\kappa} \Vert w \Vert _{2}^{2} + \frac{\kappa}{1+\kappa} \Vert \nabla w \Vert _{2}^{2} = \frac{\kappa}{1+\kappa} \Vert \nabla w \Vert _{H_{0}^{1}{( \Omega )}}^{2}. \end{aligned}$$

(4.11)

Now set $\beta _{5} = \min\{\beta _{3}, \beta _{4}\}$. Combining (4.9) and (4.10) and using (4.11), we obtain

$$\begin{aligned} \bigl(N'(t)\bigr)^{\frac{2}{p_{+}}} + \bigl(N'(t)\bigr)^{\frac{2}{p_{-}}} \geq \beta _{5} \Vert \nabla w \Vert _{2}^{2} \geq \frac{\kappa \beta _{5}}{1+\kappa} \Vert \nabla w \Vert _{H_{0}^{1}{\Omega}}^{2} = \beta _{6} N(t), \end{aligned}$$

(4.12)

where $\beta _{6} = \frac{\kappa \beta _{5}}{1+\kappa}$. Since we have the fact that $N(t) > N(0)>0$, from (4.12), we get

$$ \bigl(N'(t)\bigr)^{\frac{2}{p_{+}}} \geq \frac{\beta _{6}}{2}N(0) \quad \text{or} \quad \bigl(N'(t)\bigr)^{\frac{2}{p_{-}}} \geq \frac{\beta _{6}}{2}N(0). $$

Consequently,

$$ \bigl(N'(t)\bigr) \geq \frac{\beta _{7}}{2}N(0)^{\frac{p_{+}}{2}} \quad \text{or} \quad \bigl(N'(t)\bigr) \geq \frac{\beta _{8}}{2}N(0)^{\frac{p_{-}}{2}}, $$

where $\beta _{7} = (\frac{\beta _{6}}{2})^{\frac{p_{+}}{2}}$ and $\beta _{8} = (\frac{\beta _{6}}{2})^{\frac{p_{-}}{2}}$. Now put $\beta _{9} = \operatorname{min} \{ \frac{\beta _{7}}{2}N(0)^{\frac{p_{+}}{2}}, \frac{\beta _{8}}{2}N(0)^{\frac{p_{-}}{2}} \}$, then we get

$$ \bigl(N'(t)\bigr) \geq \beta _{9}. $$

(4.13)

(4.12) implies that

$$ \bigl(N'(t) \bigr)^{\frac{2}{p_{-}}} \bigl( 1+ \bigl(N'(t) \bigr)^{ (\frac{2}{p_{+}}-\frac{2}{p_{-}} )} \bigr) \geq \beta _{6} N(t). $$

(4.14)

From (1.7), we observe that $\frac{2}{p_{+}}-\frac{2}{p_{-}}\leq 0$. Making use of (4.13) consequently, we get

$$ \bigl(N'(t)\bigr) \geq \theta \bigl( N(t) \bigr)^{\frac{p_{-}}{2}}, $$

(4.15)

where the constant $\theta = ( \frac{\beta _{6}}{1+ \beta _{9}^{ (\frac{2}{p_{+}}-\frac{2}{p_{-}} )}} )^{\frac{p_{-}}{2}}$. Integrating from 0 to t, (4.15) gives

$$ N(t)\geq \frac{1}{ [ (N(0) )^{ ( 1- \frac{p_{-}}{2} )} + (\frac{2- p_{-}}{2} )\theta t ]^{\frac{2}{p_{-} -2}}}. $$

(4.16)

This gives the finite time blow-up of the solution w at $T^{\star}$ with

$$ T^{\star}\leq \frac{2 [N(0) ]^{ (\frac{2-p_{-}}{2} )}}{(p_{-} -2)\theta}. $$

(4.17)

Hence the proof. □

5 Lower bound for blow-up time

Here we obtain a lower bound for the blow-up time of the solutions of (1.6).

Theorem 5.1

If the weak solution w of the problem (1.6) blows up at finite time $T^{\star}$, then $T^{\star}$ has a lower bound given by

$$ T^{\star}\geq \int _{N(0)}^{\infty} \frac{d\sigma}{2\alpha _{2}^{s_{-}}(\sigma )^{\frac{s_{-}}{2}}+ 2\alpha _{3}^{s_{+}}(\sigma )^{\frac{s_{+}}{2}}+2C\alpha _{1}(\sigma )^{\frac{h+\eta}{2}}}, $$

(5.1)

where C, $\alpha _{1}$, $\alpha _{2}$ and $\alpha _{3}$ are constants.

Proof

Consider $N(t)$ as in (4.5). From the previous section, we have

$$\begin{aligned} N'(t) =& 2 \biggl[ \int _{\Omega} \vert w \vert ^{s(x)} \,dx - \int _{\Omega} \vert \nabla w \vert ^{p(x)} \,dx + \int _{\Omega} \vert w \vert ^{h} \log \vert w \vert \,dx \biggr] \end{aligned}$$

(5.2)

$$\begin{aligned} \leq & 2 \biggl[ \int _{\Omega} \vert w \vert ^{s(x)} \,dx + \int _{\Omega} \vert w \vert ^{h} \log \vert w \vert \,dx \biggr]. \end{aligned}$$

(5.3)

Since we have $w^{-\eta} \operatorname{log} w \leq (e\eta )^{-1}$ for all $\eta >0$ and $w \geq 1$, we can deduce

$$\begin{aligned} \int _{\Omega} \vert w \vert ^{h} \log \vert w \vert \,dx \leq & \int _{\{x\in \Omega : \vert w \vert \geq 1 \}} \vert w \vert ^{h} \log \vert w \vert \,dx \\ \leq & ( e\eta )^{-1} \int _{\{x\in \Omega : \vert w \vert \geq 1 \}} \vert w \vert ^{h+ \eta} \,dx \leq C \Vert w \Vert _{h+\eta}^{h+\eta}\leq C \alpha _{1} \Vert \nabla w \Vert _{2}^{h+\eta} \end{aligned}$$

(5.4)

using Sobolev embedding theorem, where $\alpha _{1}$ is the embedding constant.

Thus,

$$ \int _{\Omega} \vert w \vert ^{h} \log \vert w \vert \,dx \leq C\alpha _{1} \Vert \nabla w \Vert _{2}^{h+ \eta}. $$

(5.5)

Now by Sobolev embedding theorem,

$$\begin{aligned} \int _{\Omega} \vert w \vert ^{s(x)}\,dx \leq& \int _{\Omega} \vert w \vert ^{s_{-}}\,dx + \int _{\Omega} \vert w \vert ^{s_{+}}\,dx \\ \leq& \alpha _{2}^{s_{-}} \biggl( \int _{\Omega} \vert \nabla w \vert ^{2} \,dx \biggr)^{\frac{s_{-}}{2}} + \alpha _{3}^{s_{+}} \biggl( \int _{\Omega} \vert \nabla w \vert ^{2} \,dx \biggr)^{\frac{s_{+}}{2}}, \end{aligned}$$

(5.6)

where $\alpha _{2}$ and $\alpha _{3}$ are the corresponding embedding constants. The inequalities (5.5) and (5.6) together imply

$$\begin{aligned} N'(t) \leq & 2\alpha _{2}^{s_{-}} \biggl( \int _{\Omega} \vert \nabla w \vert ^{2} \,dx \biggr)^{\frac{s_{-}}{2}} + 2\alpha _{3}^{s_{+}} \biggl( \int _{\Omega} \vert \nabla w \vert ^{2} \,dx \biggr)^{\frac{s_{+}}{2}} + 2 C\alpha _{1} \biggl( \int _{\Omega} \vert \nabla w \vert ^{2}\,dx \biggr)^{\frac{h+\eta}{2}} \\ \leq & 2\alpha _{2}^{s_{-}} \bigl(N(t) \bigr)^{\frac{s_{-}}{2}} + 2 \alpha _{3}^{s_{+}} \bigl(N(t) \bigr)^{\frac{s_{+}}{2}} + 2 C\alpha _{1} \bigl(N(t) \bigr)^{\frac{h+\eta}{2}}. \end{aligned}$$

(5.7)

Integrating (5.7) from 0 to t, we get

$$ \int _{N(0)}^{N(t)} \frac{d\sigma}{2\alpha _{2}^{s_{-}}(\sigma )^{\frac{s_{-}}{2}}+ 2\alpha _{3}^{s_{+}}(\sigma )^{\frac{s_{+}}{2}}+2C\alpha _{1}(\sigma )^{\frac{h+\eta}{2}}} \leq t. $$

(5.8)

Theorem 4.1 ensures the existence of finite time blow-up. Thus, from (5.8), we get a lower bound as below

$$ T^{\star}\geq \int _{N(0)}^{\infty} \frac{d\sigma}{2\alpha _{2}^{s_{-}}(\sigma )^{\frac{s_{-}}{2}}+ 2\alpha _{3}^{s_{+}}(\sigma )^{\frac{s_{+}}{2}}+2C\alpha _{1}(\sigma )^{\frac{h+\eta}{2}}}, $$

(5.9)

which completes the proof. □

6 Conclusion

In history, there are many studies devoted to logarithmic nonlinearity or polynomial nonlinearity. The work in this paper is about what happens to the solutions when we combine these two nonlinearities together. Here we established the existence and finite time blow-up of solutions for the case when $s(x) < h$. Also, we obtained upper and lower bounds for the blow-up time under suitable conditions. The case $s(x)> h$ is still under study.