Skip to main content

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.

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, 1719, 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.

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.

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. □

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. □

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. □

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.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Barrow, J., Parsons, P.: In inflationary models with logarithmic potentials. Phys. Rev. D 52, 5576–5587 (1995)

    Article  Google Scholar 

  2. Bartkowski, K., Gorka, P.: One-dimensional Klein-Gordon equation with logarithmic non-linearities. J. Phys. A 41, 355201 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  3. Bernini, F., Mugnai, D.: On a logarithmic Hartree equation. Adv. Nonlinear Anal. 9, 850–865 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  4. Bialynicki-Birula, I., Mycielski, J.: Wave equations with logarithmic nonlinearities. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 23, 461–466 (1975)

    MathSciNet  Google Scholar 

  5. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    MATH  Book  Google Scholar 

  6. Cao, Y., Liu, C.: Initial boundary value problem for a mixed pseudo parabolic p-Laplacian type equation with logarithmic nonlinearity. Electron. J. Differ. Equ. 116, 1 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Chen, H., Luo, P., Liu, G.: Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity. J. Math. Anal. Appl. 422, 84–98 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  8. Chen, H., Tian, S.: Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J. Differ. Equ. 258, 4424–4442 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  9. Dai, P., Mu, C., Xu, G.: Blow-up phenomena for a pseudo-parabolic equation with p-Laplacian and logarithmic nonlinearity terms. J. Math. Anal. Appl. 481, 123439 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  10. Di, H., Shang, Y., Peng, X.: Blow-up phenomena for a pseudo-parabolic equation with variable exponents. Appl. Math. Lett. 64, 67–73 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  11. Diening, L., Hästö, P., Harjulehto, P., Ružička, M.M.: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Berlin (2011)

    MATH  Book  Google Scholar 

  12. Ding, H., Zhou, J.: Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity. J. Math. Anal. Appl. 478, 393–420 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  13. Enqvist, K., McDonald, J.: Q-balls and baryogenesis in the MSSM. Phys. Lett. B 425, 309–321 (1998)

    Article  Google Scholar 

  14. Fan, X., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m,p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  15. Gao, Y., Guo, B., Gao, W.: Weak solutions for a high-order pseudo-parabolic equation with variable exponents. Appl. Anal. 93, 322–338 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  16. Gorka, P.: Logarithmic Klein-Gordon equation. Acta Phys. Pol. B 40, 59–66 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Gurusamy, A., Balachandran, K.: Blow-up of solutions to reaction-diffusion system with nonstandard growth conditions. J. Appl. Nonlinear Dyn. 6, 407–425 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  18. Gurusamy, A., Balachandran, K.: Global existence and blow-up of solutions to a coupled parabolic system. Indian J. Ind. Appl. Math. 11, 138–156 (2020)

    Article  Google Scholar 

  19. Gurusamy, A., Shangerganesh, L., Balachandran, K.: Blow-up solutions of nonlinear parabolic system with variable exponents. Nonlinear Funct. Anal. Appl. 21, 449–461 (2016)

    MATH  Google Scholar 

  20. Han, Y.: Finite time blowup for a semilinear pseudo-parabolic equation with general nonlinearity. Appl. Math. Lett. 99, 105986 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  21. Han, Y., Cao, C., Sun, P.: A p p-Laplace equation with logarithmic nonlinearity at high initial energy level. Acta Appl. Math. 164, 155–164 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  22. He, Y., Gao, H., Wang, H.: Blow-up and decay for a class of pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity. Comput. Math. Appl. 75, 459–469 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  23. Huan, Y.: Blow-up at infinity of solutions to a semilinear heat equation with logarithmic nonlinearity. J. Math. Anal. Appl. 474, 513–517 (2019)

    MathSciNet  Article  Google Scholar 

  24. Korpusov, M.O., Sveshnikov, A.G.: Three dimensional non-linear evolutionary pseudo - parabolic equations in mathematical physics. Ž. Vyčisl. Mat. Mat. Fiz. 43, 1835–1869 (2003)

    MATH  Google Scholar 

  25. Lian, W., Ahmed, M.S., Xu, R.: Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity. Opusc. Math. 40, 111–130 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  26. Lian, W., Wang, J., Xu, R.: Global existence and blow up of solutions for pseudo-parabolic equation with singular potential. J. Differ. Equ. 269, 4914–4959 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  27. Lian, W., Xu, R.: Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. 9, 613–632 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  28. Liao, M.: Non-global existence of solutions to pseudo-parabolic equations with variable exponents and positive initial energy. C. R. Mecanique 347, 710–715 (2019)

    Article  Google Scholar 

  29. Liao, M., Guo, B., Li, Q.: Global existence and energy decay estimates for weak solutions to the pseudo-parabolic equation with variable exponents. Math. Methods Appl. Sci. 43, 2516–2527 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  30. Luo, P.: Blow-up phenomena for a pseudo-parabolic equation. Math. Methods Appl. Sci. 38, 2636–2641 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  31. Marras, M., Vernier-Piro, S., Viglialoro, G.: Blow-up phenomena for nonlinear pseudo parabolic equations with gradient term. Discrete Contin. Dyn. Syst., Ser. B 22, 2291–2300 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Meyvaci, M.: Blow up of solutions of pseudo parabolic equations. J. Math. Anal. Appl. 352, 629–633 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  33. Meyvaci, M.: Bounds for blow-up time in nonlinear pseudo-parabolic equations. Mediterr. J. Math. 15, Article ID 8 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  34. Nhan, L.C., Truong, L.X.: Global solution and blow-up for a class of pseudo p-Laplacian evolution equations with logarithmic nonlinearity. Comput. Math. Appl. 73, 2076–2091 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  35. Nhan, L.C., Truong, L.X.: Global solution and blow-up for a class of p-Laplacian evolution equations with logarithmic nonlinearity. Acta Appl. Math. 151(1), 149–169 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  36. Peng, X., Shang, Y., Zheng, X.: Blow-up phenomena for some nonlinear pseudo-parabolic equations. Appl. Math. Lett. 56, 17–22 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  37. Sarra, T., Zara, A., Boulaaras, S.: Decay estimate and non-extinction of solutions of p-Laplacian nonlocal heat equations. AIMS Math. 5, 1663–1679 (2020)

    MathSciNet  Article  Google Scholar 

  38. Shangerganesh, L., Gurusamy, A., Balachandran, K.: Weak solutions for nonlinear parabolic equations with variable exponents. Commun. Math. 25, 55–70 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  39. Showalter, R.E., Ting, T.W.: Pseudoparabolic partial differential equations. SIAM J. Math. Anal. 1, 1–26 (1970)

    MathSciNet  MATH  Article  Google Scholar 

  40. Simon, J.: Compact sets in the space \(L^{p}(0, T; B)\). Annali di Matematica Pura et Applicanta 146, 65–96 (1987)

    MATH  Article  Google Scholar 

  41. Sun, F., Liu, L., Wu, Y.: Finite time blow-up for a class of parabolic or pseudo-parabolic equations. Comput. Math. Appl. 75, 3685–3701 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  42. Wang, X., Chen, Y., Yang, Y., Li, J., Xu, R.: Kirchhoff-type system with linear weak damping and logarithmic nonlinearities. Nonlinear Anal. 188, 475–499 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  43. Wang, X., Xu, R.: Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation. Adv. Nonlinear Anal. 10, 261–288 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  44. Xu, R., Su, J.: Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. 264, 2732–2763 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  45. Xu, R., Wang, X., Yang, Y.: Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy. Appl. Math. Lett. 83, 176–181 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  46. Zhu, X., Li, F., Li, Y.: Some sharp results about the global existence and blowup of solutions to a class of pseudo-parabolic equations. Proc. R. Soc. Edinb. 147A, 1311–1331 (2017)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the referee for the improvement of the paper.

Funding

The first author would like to thank the Department of Science and Technology, New Delhi, for their financial support under the INSPIRE fellowship program. This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2021R1F1A1048937) to the last author.

Author information

Authors and Affiliations

Authors

Contributions

All the authors have contributed equally to this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yong-Ki Ma.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lakshmipriya, N., Gnanavel, S., Balachandran, K. et al. Existence and blow-up of weak solutions of a pseudo-parabolic equation with logarithmic nonlinearity. Bound Value Probl 2022, 30 (2022). https://doi.org/10.1186/s13661-022-01611-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13661-022-01611-3

MSC

  • 35D30
  • 35B44
  • 35K70

Keywords

  • Weak solutions
  • Blow-up
  • Variable exponent spaces
  • Galerkin method
  • \(p(x)\)-Laplacian operator