Blow-up and lifespan of solutions for elastic membrane equation with distributed delay and logarithmic nonlinearity

Abstract

We examine a Kirchhoff-type equation with nonlinear viscoelastic properties, characterized by distributed delay, logarithmic nonlinearity, and Balakrishnan–Taylor damping terms (elastic membrane equation). Under appropriate hypotheses, we establish the occurrence of solution blow-up.

1 Introduction

Analyzing nonlinear mathematical problems involves a distinct set of challenges and techniques compared to linear problems [1, 2]. Nonlinear problems often manifest in various scientific and engineering domains, and their analysis is crucial for gaining insights into complex phenomena [3, 4]. The systematic examination of mathematical problems necessitates a methodical approach encompassing rigorous formulation, assessment of existence and uniqueness of solutions, linearization procedures, stability analyses, application of numerical methodologies, bifurcation investigations, phase plane analyses, sensitivity assessments, optimization strategies, and the validation and verification of outcomes [5, 6]. This multifaceted framework is imperative for elucidating the intricate dynamics inherent in various systems across diverse scientific and engineering domains [7].

The analysis of solutions to the Kirchhoff equation concerning viscoelastic materials is paramount in comprehending the mechanical characteristics of such materials. These solutions serve as a cornerstone in directing the design methodology, thereby ensuring the dependability and efficacy of materials across diverse applications. In this current study, we examine the Kirchhoff equation provided below:

$$ \textstyle\begin{cases} v_{tt}-N(t)\Delta v(t)+\int _{0}^{t}f(t-\Lambda ) \Delta v(\Lambda )\,d\Lambda +\mu _{1}v_{t} +\int _{\tau _{1}}^{ \tau _{2}}\mu _{2} (s) v_{t}(y,t-s) \,ds \\ \quad =v \vert v \vert ^{\gamma -2}\ln \vert v \vert ^{l}, \quad x\in \Omega , 0< t, \\ v( y,0) =v_{0}( y), \qquad v_{t}( y,0) =v_{1}( y), \quad y\in \Omega , \\ v_{t}( y,-t) =h_{0}( y,t), \quad y\in \Omega , t\in (0, \tau _{2}), \\ v( y,t) =0, \quad (y,t)\in \partial \Omega \times (0, \infty ), \end{cases} $$

(1.1)

where

$$ N(t):= \bigl(\zeta _{0}+\zeta _{1} \Vert \nabla v \Vert ^{2}_{2}+\sigma \bigl( \nabla v(t),\nabla v_{t}(t)\bigr)_{M^{2}(\Omega )} \bigr). $$

In this expression, Ω belongs to the set of bounded domains in $\mathbb{R}^{N}$ and possesses a boundary ∂Ω that is suitably smooth. $\gamma \geq 2$, $\zeta _{0}$, $\zeta _{1}$, σ, $\mu _{1}$, l are positive constants. In addition to this, the time delays are indicated by $\tau _{1}$, $\tau _{2}$ with $0\leq \tau _{1}<\tau _{2}$, while $\mu _{2}$ is an $M^{\infty}$ function and f is a positive function.

In a physical sense, the connection between the stress and strain history in the beam is influenced by a viscoelastic damping term inspired by Boltzmann theory. The kernel of memory term in this context is represented by the function f, which is frequently discussed in the literature [8–16].

In [17], Balakrishnan and Taylor introduced a novel damping model known as Balakrishnan–Taylor damping, specifically addressing concerns related to the span problem and the plate equation. Numerous studies have explored this damping phenomenon, as documented in [11, 14, 15, 17–20, 35–37], and [21]. The occurrence of delay is a common feature in various applications and practical problems, rendering many systems worthy of investigation. Recently, several authors have directed their attention towards analyzing the asymptotic behavior and stability of evolution systems incorporating time delay, as discussed in the research [9–12, 15, 22–26], and [27].

The significance of logarithmic nonlinearity in physical system is emphasized by its involvement in a wide range of topics and theories, encompassing symmetry, cosmology, quantum mechanics, nuclear physics, and various applications including nuclear, optical, and subterranean physics. Different authors have delved into such type of problem across various domains, exploring aspects such as global solution existence, stability, blow-up, and the growth of solutions, as documented in works such as [11, 19, 28–31], and [32–34]. Taking into account various elements, damping terms including the distributed delay terms, logarithmic nonlinearity, Balakrishnan–Taylor damping, and memory term are integrated into a specific problem, along with the incorporation of $\int _{\tau {1}}^{\tau {2}}\mu{2} (s)v{t}(y,t-s) \,ds$. Further investigation is required to explore such type of novel and distinctive problem. This diverges from the previously mentioned scenarios, and our objective is to shed light on this unique problem.

Our work is structured as follows: In the subsequent section, we lay out the necessary lemmas, concepts, and hypotheses. In Sect. 3, we state and prove the main blow-up solution results. We present the concluding remarks of our work in Sect. 4 of this work.

2 Fundamental concepts

Here, to investigate our problem, we require certain materials. To begin with, we present the following assumptions regarding $\beta _{2}$ and f:

(H1):

$f:\mathbb{R}{+}\rightarrow \mathbb{R}{+}$ represents nonincreasing $B^{1}$ functions fulfilling

$$ 0< f(t), \quad \zeta _{0}- \int _{0}^{\infty}f( \Lambda ) \,d\Lambda = l>0. $$

(2.1)

(H2):

$\mu _{2}:[\tau _{1},\tau _{2}]\rightarrow \mathbb{R}$ is an $M^{\infty}$ function in a way that

$$ \biggl(\frac{2\delta +1}{2}\biggr) \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert \,ds< \mu _{1} , \quad \delta >\frac{1}{2}, $$

(2.2)

with

$$ (f\circ \psi ) (t):= \int _{\Omega} \int _{0}^{t}f(t-\Lambda ) \bigl\vert \psi (t)- \psi (\Lambda ) \bigr\vert ^{2}\,d\Lambda \,dy. $$

As in [27], we take the following:

$$ x(y,\rho ,s,t)=v_{t}(y,t-s\rho ), \quad (y,\rho ,s,t)\in \Omega \times (0, 1)\times (\tau _{1},\tau _{2}) \times \mathbb{R}_{+}, $$

which satisfy

$$ \textstyle\begin{cases} sx_{t}(y,\rho ,s,t)+x_{\rho}(y,\rho ,s,t)=0, \\ x(y,0,s,t)=v_{t}(y,t). \end{cases} $$

(2.3)

Then one can write (1.1) as follows:

$$ \textstyle\begin{cases} v_{tt}-N(t)\Delta v(t)+\int _{0}^{t}f(t-\Lambda ) \Delta v(\Lambda )\,d\Lambda +\mu _{1}v_{t}+\int _{\tau _{1}}^{ \tau _{2}}\mu _{2} (s)x(y,1,s,t) \,ds \\ \quad =v \vert v \vert ^{\gamma -2}\ln \vert v \vert ^{l}, \\ sx_{t}(y,\rho ,s,t)+x_{\rho}(y,\rho ,s,t)=0, \\ v(y,0)=v_{0}( y), \quad v_{t}( y,0)=v_{1}( y), \quad y\in \Omega , \\ x(y,\rho ,s,0)=h_{0}(y,s\rho ), \quad \text{in} \ \Omega \times (0,1)\times (\tau _{1},\tau _{2}), \\ v(y,t)=0, \quad (x,t)\in \partial \Omega \times (0, \infty ). \end{cases} $$

(2.4)

Here, the energy functional is introduced as follows.

Lemma 2.1

Let Q represent the energy functional given by

$$\begin{aligned} Q( t) =&\frac{1}{2} \Vert v_{t} \Vert _{2}^{2}+ \frac{1}{2} \biggl(\zeta _{0}- \int _{0}^{t}v(\Lambda )\,d\Lambda \biggr) \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{2}+ \frac{\zeta _{1}}{4} \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{4} \\ &{}+\frac{1}{2}(f\circ \nabla v) (t)+\frac{l}{\gamma} \bigl\Vert v(t) \bigr\Vert _{ \gamma}^{\gamma}-\frac{1}{\gamma} \int _{\Omega} \vert v \vert ^{\gamma} \ln \vert v \vert ^{l} \,dy \\ &{}+\frac{1}{2} \int _{\Omega} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}s \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,\rho ,s,t)\,ds \,d\rho \,dy, \end{aligned}$$

(2.5)

which satisfies

$$\begin{aligned} Q^{\prime } ( t ) \leq &-B_{0} \biggl( \Vert v_{t} \Vert _{2}^{2}+ \int _{\Omega} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,1,s,t)\,ds \,dx \biggr)+\frac{1}{2} \bigl(f^{\prime }\circ \nabla v\bigr) (t) \\ &{}-\frac{1}{2}f(t) \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2}-\frac{\sigma}{4} \biggl(\frac{d}{dt} \bigl\{ \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2} \bigr\} \biggr)^{2} \\ \leq& 0. \end{aligned}$$

(2.6)

Proof

By taking the inner product of (2.4)₁ with vt and subsequently integrating over Ω, we obtain

$$\begin{aligned}& \bigl(v_{tt}(t),v_{t}(t)\bigr)_{M^{2}(\Omega )}-\bigl(N(t) \Delta v(t),v_{t}(t)\bigr)_{M^{2}( \Omega )} \\& \quad {}+\biggl( \int _{0}^{t}f(t-\Lambda )\Delta v(\Lambda )\,d\Lambda ,v_{t}(t)\biggr)_{M^{2}( \Omega )}+\mu _{1}(v_{t},v_{t})_{M^{2}(\Omega )} \\& \quad {}+\biggl( \int _{\tau _{1}}^{\tau _{2}}\mu _{2} (s)x(y,1,s,t) \,ds,v_{t}(t)\biggr)_{M^{2}( \Omega )}-\bigl(lv \vert v \vert ^{\gamma -2}\ln \vert v \vert , v_{t}(t)\bigr)_{M^{2}( \Omega )}=0. \end{aligned}$$

(2.7)

Then a straightforward calculation yields

$$\begin{aligned} \bigl(v_{tt}(t),v_{t}(t)\bigr)_{M^{2}(\Omega )} =& \frac {1}{2}\frac {d}{dt} \bigl( \bigl\Vert v_{t}(t) \bigr\Vert ^{2}_{2} \bigr), \end{aligned}$$

(2.8)

further simplification implies that

$$\begin{aligned}& -\bigl(N(t)\Delta v(t),v_{t}(t)\bigr)_{M^{2}(\Omega )} \\& \quad =-\bigl( \bigl(\zeta _{0}+\zeta _{1} \Vert \nabla v \Vert ^{2}_{2}+\sigma \bigl( \nabla v(t),\nabla v_{t}(t)\bigr)_{M^{2}(\Omega )} \bigr)\Delta v(t),v_{t}(t) \bigr)_{M^{2}( \Omega )} \\& \quad = \bigl(\zeta _{0}+\zeta _{1} \Vert \nabla v \Vert ^{2}_{2}+\sigma \bigl( \nabla v(t),\nabla v_{t}(t)\bigr)_{M^{2}(\Omega )} \bigr) \int _{\Omega} \nabla v(t).\nabla v_{t}(t)\,dy \\& \quad = \bigl(\zeta _{0}+\zeta _{1} \Vert \nabla v \Vert ^{2}_{2}+\sigma \bigl( \nabla v(t),\nabla v_{t}(t)\bigr)_{M^{2}(\Omega )} \bigr)\frac{d}{dt} \biggl\{ \frac{1}{2} \int _{\Omega} \bigl\vert \nabla v(t) \bigr\vert ^{2}\,dy \biggr\} \\& \quad =\frac {d}{dt} \biggl\{ \frac{1}{2} \biggl(\zeta _{0}+ \frac{\zeta _{1}}{2} \Vert \nabla v \Vert ^{2}_{2} \biggr) \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2} \biggr\} +\frac{\sigma}{4} \biggl\{ \frac {d}{dt} \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2} \biggr\} ^{2}, \end{aligned}$$

(2.9)

with

$$\begin{aligned}& \biggl( \int _{0}^{t}f(t-\Lambda )\Delta v(\Lambda )\,d\Lambda ,v_{t}(t)\biggr)_{L^{2}( \Omega )} \\& \quad = \int _{0}^{t}f(t-\Lambda ) \bigl(\Delta v(\Lambda ),v_{t}(t)\bigr)_{M^{2}( \Omega )}\,d\Lambda \\& \quad =- \int _{0}^{t}f(t-\Lambda ) \biggl[ \int _{\Omega}\nabla v(y,\Lambda ) \nabla v(y,t)\,dy \biggr]\,d\Lambda , \end{aligned}$$

(2.10)

and

$$ -\nabla v(y,\Lambda ).\nabla v(y,t)=\frac{1}{2}\frac{d}{dt} \bigl\{ \bigl\vert \nabla v(y,\Lambda )-\nabla v(y,t) (t) \bigr\vert ^{2} \bigr\} - \frac{1}{2}\frac{d}{dt} \bigl\{ \bigl\vert \nabla v(y,t) \bigr\vert ^{2} \bigr\} . $$

(2.11)

Then we have

$$\begin{aligned}& - \int _{0}^{t}f(t-\Lambda ) \bigl(\nabla v(\varrho ), \nabla v_{t}(t)\bigr)_{M^{2}( \Omega )}\,d\Lambda \\& \quad =- \int _{0}^{t}f(t-\Lambda ) \int _{\Omega} \biggl[\frac{1}{2} \frac{d}{dt} \bigl\{ \bigl\vert \nabla v(y,\Lambda )-\nabla v(y,t) \bigr\vert ^{2} \bigr\} \biggr]\,dy \,ds. \\& \qquad {}- \int _{0}^{t}f(t-\Lambda ) \int _{\Omega} \biggl[\frac{1}{2} \frac{d}{dt} \bigl\{ \bigl\vert \nabla v(y,t) \bigr\vert ^{2} \bigr\} \biggr]\,dy \,d \Lambda \\& \quad =\frac{1}{2} \int _{0}^{t}f(t-\Lambda ) \biggl[\frac{d}{dt} \biggl\{ \int _{\Omega} \bigl\vert \nabla v(y,t)-\nabla v(y,\Lambda ) \bigr\vert ^{2}\,dy \biggr\} \biggr]\,d\Lambda \\& \qquad {}-\frac{1}{2} \int _{0}^{t}f(t-\Lambda ) \biggl[\frac{d}{dt} \bigl\{ \bigl\Vert \nabla v(y,t) \bigr\Vert _{2}^{2} \bigr\} \biggr]\,dy \,d\Lambda . \end{aligned}$$

(2.12)

Using (2.1), one has

$$\begin{aligned}& \frac{1}{2} \int _{0}^{t}f(t-\Lambda ) \biggl[\frac{d}{dt} \biggl\{ \int _{ \Omega} \bigl\vert \nabla v(y,t)-\nabla v(y,\Lambda ) \bigr\vert ^{2}\,dy \biggr\} \biggr]\,d\Lambda \\& \quad =\frac{1}{2}\frac{d}{dt} \biggl\{ \int _{0}^{t}f(t-\Lambda ) \biggl[ \int _{\Omega} \bigl\vert \nabla v(y,t)-\nabla v(y,\Lambda ) \bigr\vert ^{2}\,dy \biggr] \biggr\} \,d\Lambda \\& \qquad {}-\frac{1}{2} \int _{0}^{t}f'(t-\Lambda ) \biggl[ \int _{\Omega} \bigl\vert \nabla v(y,t)-\nabla v(y,\Lambda ) \bigr\vert ^{2}\,dy \biggr]\,d\Lambda \\& \quad =\frac{1}{2}\frac{d}{dt}(f\circ \nabla u) (t)- \frac{1}{2}\bigl(f'\circ \nabla u\bigr) (t) \end{aligned}$$

(2.13)

and

$$\begin{aligned}& -\frac{1}{2} \int _{0}^{t}h(t-\Lambda ) \biggl[\frac{d}{dt} \bigl\{ \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{2} \bigr\} \biggr]\,dy \,d\Lambda \\& \quad =-\frac{1}{2} \biggl( \int _{0}^{t}f(t-\Lambda )\,d\Lambda \biggr) \biggl( \frac{d}{dt} \bigl\{ \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{2} \bigr\} \biggr)\,dy \\& \quad =-\frac{1}{2} \biggl( \int _{0}^{t}f(\Lambda )\,d\Lambda \biggr) \biggl( \frac{d}{dt} \bigl\{ \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{2} \bigr\} \biggr)\,dy \\& \quad =-\frac{1}{2}\frac{d}{dt} \biggl\{ \biggl( \int _{0}^{t}f(\Lambda )\,d \Lambda \biggr) \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{2} \biggr\} + \frac{1}{2}f(t) \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{2}. \end{aligned}$$

(2.14)

By substituting (2.13) and (2.14) into (2.12), we have

$$\begin{aligned}& \biggl( \int _{0}^{t}f(t-\Lambda )\Delta v(\Lambda )\,d\Lambda , v_{t}(t) \biggr)_{M^{2}(\Omega )} \\& \quad =\frac{d}{dt} \biggl\{ \frac{1}{2}(f\circ \nabla v) (t) - \frac{1}{2} \biggl( \int _{0}^{t}f(\Lambda )\,d\Lambda \biggr) \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{2} \biggr\} \\& \qquad {}-\frac{1}{2}\bigl(f'\circ \nabla v\bigr) (t)+ \frac{1}{2}f(t) \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{2} \end{aligned}$$

(2.15)

and

$$\begin{aligned} -\bigl( lv \vert v \vert ^{\gamma -2}\ln \vert u \vert , v_{t}(t)\bigr)_{M^{2}( \Omega )} =\frac{d}{dt} \biggl\{ \frac{l}{\gamma} \bigl\Vert v(t) \bigr\Vert _{ \gamma}^{\gamma}- \frac{1}{\gamma} \int _{\Omega} \vert v \vert ^{\gamma} \ln \vert v \vert ^{l}\,dy \biggr\} . \end{aligned}$$

(2.16)

Here, multiply $x\vert \mu _{2}(s)\vert$ with equation (2.4)₂ by $x\vert \mu _{2}(s)\vert$ and integrate over $\Omega \times (0, 1)\times (\tau _{1},\tau _{2})$. Then applying (2.3)₂, the following is obtained:

$$\begin{aligned}& \frac{d}{dt}\frac{1}{2} \int _{\Omega} \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}}s \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,\rho ,s,t)\,ds \,d\rho \,dy \\& \quad =-\frac{1}{2} \int _{\Omega} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}2 \bigl\vert \mu _{2}(s) \bigr\vert xx_{\rho}\,ds \,d\rho \,dy \\& \quad =-\frac{1}{2} \int _{\Omega} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert \bigl[x^{2}(y,1,s,t)- x^{2}(y,0,s,t) \bigr]\,ds \,dy \\& \quad =\frac{1}{2}\biggl( \int _{\tau _{1}}^{\tau _{2}}\vert \mu _{2}(s)\,ds\biggr) \Vert u_{t} \Vert _{2}^{2}-\frac{1}{2} \int _{\Omega} \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,1,s,t)\,ds \,dy, \end{aligned}$$

(2.17)

and through the application of Young’s inequality, we obtain

$$\begin{aligned} \biggl( \int _{\tau _{1}}^{\tau _{2}}\mu _{2} (s)x(y,1,s,t) \,ds,v_{t}(t)\biggr)_{M^{2}( \Omega )}&\leq \delta \biggl( \int _{\tau _{1}}^{\tau _{2}}\vert \mu _{2}(s)\,ds\biggr) \Vert v_{t} \Vert _{2}^{2} \\ &\quad {}+\frac{1}{4\delta} \int _{\Omega} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,1,s,t)\,ds \,dy. \end{aligned}$$

(2.18)

By replacement (2.8)–(2.9) and (2.15)–(2.18) into (2.7), we find (2.5) and

$$\begin{aligned} Q^{\prime } ( t ) \leq &- \biggl(\mu _{1}-\biggl(\delta + \frac{1}{2}\biggr) \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert \,ds \biggr) \Vert v_{t} \Vert _{2}^{2} \end{aligned}$$

(2.19)

$$\begin{aligned} &{}- \biggl(\frac{2\delta -1}{4\delta} \biggr) \int _{\Omega} \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,1,s,t)\,ds \,dy+\frac{1}{2}\bigl(f^{ \prime } \circ \nabla v\bigr) (t) \\ &{}-\frac{1}{2}f(t) \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2}-\frac{\sigma}{4} \biggl(\frac{d}{dt} \bigl\{ \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2} \bigr\} \biggr)^{2} \\ \leq& 0. \end{aligned}$$

(2.20)

Thus, according to (2.2), we get(2.6), where $B_{0}>0$. Hence, the proof is completed. □

Theorem 2.2

Let us consider that (2.1)–(2.2) hold true. For any $v_{0},u_{1}\in F^{1}{0}(\Omega )\cap M^{2}(\Omega )$ and $f{0}\in M^{2}(\Omega ,(0,1))$, one can find a weak solution v to (2.4) such that

$$\begin{aligned}& v\in B\bigl(]0,P[,F^{1}_{0}(\Omega )\bigr)\cap Q^{1}\bigl(]0,P[,M^{2}(\Omega )\bigr), \\& v_{t}\in B\bigl(]0,T[,F^{1}_{0}(\Omega )\bigr)\cap M^{2}\bigl(]0,P[,M^{2}\bigl(\Omega ,(0,1) \bigr)\bigr). \end{aligned}$$

Lemma 2.3

[34] One can find a constant $b(\Omega )>0$ in a manner that

$$\begin{aligned} \biggl( \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy \biggr)^{\frac{s}{\gamma}} \leq &b \biggl( \int _{\Omega} \vert v \vert ^{ \gamma}\ln \vert v \vert ^{l} \,dy+ \Vert \nabla v \Vert _{2}^{2} \biggr) \end{aligned}$$

for any $2\leq s\leq \gamma$, provided that $0 \leq \int _{\Omega}\vert v\vert ^{\gamma}\ln \vert v\vert ^{l} \,dy$.

Corollary 2.4

[34] One can find a constant $b(\Omega )>0$ in a way that

$$\begin{aligned} \Vert v \Vert ^{2}_{2} \leq &c \biggl[ \biggl( \int _{\Omega} \vert v \vert ^{ \gamma}\ln \vert v \vert ^{l} \,dy \biggr)^{\frac{2}{\gamma}}+ \Vert \nabla v \Vert _{2}^{\frac{4}{\gamma}} \biggr], \end{aligned}$$

provided that $0 \leq \int _{\Omega}\vert v\vert ^{\gamma}\ln \vert v\vert ^{l} \,dy$.

Lemma 2.5

[34] Let us take a constant $b(\Omega )>0$ in a way that

$$\begin{aligned} \Vert v \Vert ^{s}_{\gamma} \leq &b \bigl( \Vert v \Vert ^{\gamma}_{ \gamma}+ \Vert \nabla v \Vert _{2}^{2} \bigr) \end{aligned}$$

for any $v\in M^{\gamma}(\Omega )$ and $2\leq s\leq \gamma$.

3 Blow-up result

Here, we establish the blow-up results for the solution of (2.4). First of all, the functional is introduced as

$$\begin{aligned} \mathbb{F}(t)=-Q(t) =&-\frac{1}{2} \Vert v_{t} \Vert _{2}^{2}- \frac{1}{2} \biggl(\zeta _{0}- \int _{0}^{t}f(\Lambda )\,d\Lambda \biggr) \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{2}-\frac{\zeta _{1}}{4} \bigl\Vert \nabla v(t) \bigr\Vert _{2}^{4} \\ &{}-\frac{1}{2}(f\circ \nabla v) (t)-\frac{l}{\gamma} \bigl\Vert v(t) \bigr\Vert _{ \gamma}^{\gamma}+\frac{1}{\gamma} \int _{\Omega} \vert v \vert ^{\gamma} \ln \vert v \vert ^{l} \,dy \\ &{}-\frac{1}{2} \int _{\Omega} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}s \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,\rho ,s,t)\,ds \,d\rho \,dy. \end{aligned}$$

(3.1)

Theorem 3.1

Assuming that (2.1)–(2.2) are satisfied, and given that $Q(0)<0$, the solution to problem (2.4) experiences a finite time blow-up.

Proof

For the required proof, the following is obtained from (2.6):

$$ Q(t)\leq Q(0)\leq 0; $$

(3.2)

thus, we have

$$\begin{aligned} \mathbb{F}'(t)=-Q'(t) \geq &B_{0} \biggl( \Vert v_{t} \Vert _{2}^{2}+ \int _{\Omega} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,1,s,t)\,ds \,dy \biggr), \end{aligned}$$

(3.3)

which implies that

$$\begin{aligned} \mathbb{F}'(t) \geq &B_{0} \bigl\Vert v_{t}(t) \bigr\Vert ^{2}_{2}\geq 0 \\ \mathbb{F}'(t) \geq &B_{0} \int _{\Omega} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,1,s,t)\,ds \,dy\geq 0 . \end{aligned}$$

(3.4)

By (3.1), we have

$$\begin{aligned} 0\leq \mathbb{F}(0)\leq \mathbb{F}(t) \leq &\frac{1}{\gamma} \int _{ \Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy . \end{aligned}$$

(3.5)

We set

$$\begin{aligned} \mathcal{L}(t)=\mathbb{F}^{1-\alpha}(t)+\varepsilon \int _{\Omega}vv_{t}\,dy+ \frac{\varepsilon \mu _{1}}{2} \int _{\Omega}v^{2}\,dy +\frac{\sigma}{4} \Vert \nabla v \Vert ^{4}_{2}, \end{aligned}$$

(3.6)

where $\varepsilon >0$ will be assigned a specific value later, and

$$\begin{aligned} \frac{2(\gamma -1)}{\gamma ^{2}}< \alpha < \frac {\gamma -2}{2\gamma}< 1. \end{aligned}$$

(3.7)

Multiplying v with (2.4)₁ and taking the derivative of (3.6), the following is obtained:

$$\begin{aligned} \mathcal{L}'(t) = &(1-\alpha )\mathbb{F}^{-\alpha} \mathbb{F}'(t)+ \varepsilon \Vert v_{t} \Vert _{2}^{2}+\varepsilon \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy \\ &{}-\varepsilon \zeta _{0} \Vert \nabla v \Vert _{2}^{2}-\varepsilon \zeta _{1} \Vert \nabla v \Vert _{2}^{4}+ \underbrace{\varepsilon \int _{\Omega}\nabla v \int ^{t}_{0}f(t-\Lambda ) \nabla v(\Lambda )\,d \varrho \,dy}_{J_{1}} \\ &{}- \underbrace{\varepsilon \int _{\Omega} \int _{\tau _{1}}^{\tau _{2}}\mu _{2}(s)u x(y,1,s,t)\,ds \,dy}_{J_{2}}. \end{aligned}$$

(3.8)

Next, we have

$$\begin{aligned} J_{1} =&\varepsilon \int _{0}^{t}h(t-\Lambda )\,d\Lambda \int _{\Omega} \nabla v.\bigl(\nabla v(\Lambda )-\nabla v(t) \bigr)\,dy \,d\Lambda +\varepsilon \int _{0}^{t}f(\Lambda )\,d\Lambda \Vert \nabla v \Vert _{2}^{2} \\ \geq & \frac{\varepsilon}{2} \biggl( \int _{0}^{t}f(\Lambda )\,d\Lambda \biggr) \Vert \nabla v \Vert _{2}^{2}-\frac{\varepsilon}{2}(f \circ \nabla v) \end{aligned}$$

(3.9)

and, for $\delta _{1}>0$,

$$\begin{aligned} J_{2}\geq - \varepsilon \mu _{1}\delta _{1} \Vert u \Vert _{2}^{2}- \frac{\varepsilon}{4\delta _{1}} \int _{\Omega} \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,1,s,t)\,ds \,dy. \end{aligned}$$

(3.10)

From (3.8), we find

$$\begin{aligned} \mathcal{L}'(t) \geq &(1-\alpha )\mathbb{F}^{-\alpha} \mathbb{F}'(t)+ \varepsilon \Vert v_{t} \Vert _{2}^{2}+\varepsilon \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy \\ &{}-\varepsilon \zeta _{1} \Vert \nabla v \Vert _{2}^{4}-\varepsilon [ \biggl(\zeta _{0}- \frac{1}{2} \int _{0}^{t}f(\Lambda )\,d\Lambda \biggr) \Vert \nabla v \Vert _{2}^{2}-\frac{\varepsilon}{2}(f\circ \nabla v) \\ &{}- \varepsilon \mu _{1}\delta _{1} \Vert v \Vert _{2}^{2}- \frac{\varepsilon}{4\delta _{1}} \int _{\Omega} \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,1,s,t)\,ds \,dy. \end{aligned}$$

(3.11)

At this point, by setting $\delta _{1}$ so that, for large κ to be specified later

$$ \frac{1}{4\delta _{1}B_{0}}=\kappa \mathbb{F}^{-\alpha}(t), $$

by (3.4) and putting in (3.11), we get

$$\begin{aligned} \mathcal{L}'(t) \geq &\bigl[(1-\alpha )-\varepsilon \kappa \bigr] \mathbb{F}^{- \alpha}\mathbb{F}'(t)+\varepsilon \Vert v_{t} \Vert _{2}^{2} \\ &{}-\frac{\varepsilon}{2}(f\circ \nabla v)-\varepsilon \zeta _{1} \Vert \nabla v \Vert _{2}^{4}-\varepsilon \biggl(\zeta _{0}-\frac{1}{2} \int _{0}^{t}f(\Lambda )\,d\Lambda \biggr) \Vert \nabla v \Vert _{2}^{2} \\ &{}- \varepsilon \biggl( \frac{\mu _{1}\mathbb{F}^{\alpha}(t)}{4B_{0}\kappa} \biggr) \Vert v \Vert _{2}^{2}+\varepsilon \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy . \end{aligned}$$

(3.12)

Now, for $0< a<1$ and from (3.1), we have

$$\begin{aligned} \varepsilon \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy =&\varepsilon a \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy +\frac{\varepsilon \gamma (1-a)}{2} \Vert v_{t} \Vert _{2}^{2}+ \varepsilon \gamma (1-a)\mathbb{F}(t) \\ &{}+\varepsilon \frac{\gamma (1-a)}{2}\biggl(\zeta _{0}- \int _{0}^{t}f( \Lambda )\,d\Lambda \biggr) \Vert \nabla v \Vert _{2}^{2}+\varepsilon l(1-a) \Vert v \Vert _{\gamma}^{\gamma} \\ &{}+\varepsilon \frac{\zeta _{1}\gamma (1-a)}{2} \Vert \nabla v \Vert _{2}^{4}- \varepsilon \frac{\gamma (1-a)}{2}((f\circ \nabla v) \\ &{}+\frac{\varepsilon \gamma (1-a)}{2} \int _{\Omega} \int _{0}^{1} \int _{ \tau _{1}}^{\tau _{2}}s \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,\rho ,s,t)\,ds \,d \rho \,dy. \end{aligned}$$

(3.13)

Putting in (3.12), one has

$$\begin{aligned} \mathcal{L}'(t) \geq & \bigl\{ (1-\alpha )-\varepsilon \kappa \bigr\} \mathbb{F}^{-\alpha}\mathbb{F}'(t)+\varepsilon a \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert u \vert ^{l} \,dy \\ &{}+\varepsilon \biggl\{ \frac{\gamma (1-a)}{2}+1 \biggr\} \Vert v_{t} \Vert _{2}^{2}- \varepsilon \biggl( \frac{\mu _{1}\mathbb{F}^{\alpha}(t)}{4B_{0}\kappa} \biggr) \Vert v \Vert _{2}^{2} \\ &{}+\varepsilon \biggl\{ \frac{\gamma (1-a)}{2} \biggl(\zeta _{0}- \int _{0}^{t}f( \Lambda )\,d\Lambda \biggr)- \biggl( \zeta _{0}-\frac{1}{2} \int _{0}^{t}f( \Lambda )\,d\Lambda \biggr) \biggr\} \Vert \nabla v \Vert _{2}^{2} \\ &{}+\varepsilon \zeta _{1} \biggl\{ \frac{\gamma (1-a)}{2}-1 \biggr\} \Vert \nabla v \Vert _{2}^{4}+\varepsilon \biggl\{ \frac{\gamma (1-a)}{2}- \frac{1}{2} \biggr\} (f\circ \nabla v) \\ &{}+\varepsilon l(1-a) \Vert v \Vert _{\gamma}^{\gamma}+ \varepsilon \gamma (1-a)\mathbb{F}(t) \\ &{}+\frac{\varepsilon \gamma (1-a)}{2} \int _{\Omega} \int _{0}^{1} \int _{ \tau _{1}}^{\tau _{2}}s \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,\rho ,s,t)\,ds \,d \rho \,dy. \end{aligned}$$

(3.14)

According to (3.5), Corollary 2.4, and Young’s inequality, we get

$$\begin{aligned} \mathbb{F}^{\alpha}(t) \Vert v \Vert _{2}^{2} \leq & \biggl( \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy \biggr)^{\alpha} \Vert v \Vert _{2}^{2} \\ \leq &c \biggl[ \biggl( \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dx \biggr)^{\alpha +\frac{2}{\gamma}}+ \biggl( \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy \biggr)^{\alpha} \Vert \nabla v \Vert _{2}^{\frac{4}{\gamma}} \biggr] \\ \leq &b \biggl[ \biggl( \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy \biggr)^{\frac{(\alpha \gamma +2)}{\gamma}}+ \biggl( \int _{ \Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy \biggr)^{ \frac{\alpha \gamma}{(\gamma -2)}}+ \Vert \nabla v \Vert _{2}^{2} \biggr]. \end{aligned}$$

(3.7) yields

$$ 2< \alpha \gamma +2\leq \gamma \quad \text{and} \quad 2< \frac{\alpha \gamma ^{2}}{\gamma -2}\leq \gamma . $$

Hence, Lemma 2.3 gives

$$\begin{aligned} \mathbb{F}^{\alpha}(t) \Vert v \Vert _{2}^{2} \leq &c \biggl( \int _{ \Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy+ \Vert \nabla v \Vert _{2}^{2} \biggr). \end{aligned}$$

(3.15)

From (3.14) and (3.15), we have

$$\begin{aligned} \mathcal{L}'(t) \geq & \bigl\{ (1-\alpha )-\varepsilon \kappa \bigr\} \mathbb{F}^{-\alpha}\mathbb{F}'(t) +\varepsilon \biggl(a- \frac{b\mu _{1}}{4B_{0}\kappa} \biggr) \int _{\Omega} \vert u \vert ^{ \gamma}\ln \vert v \vert ^{l} \,dy \\ &{}+\varepsilon \biggl\{ \frac{\gamma (1-a)}{2}+1 \biggr\} \Vert v_{t} \Vert _{2}^{2}+\varepsilon l(1-a) \Vert v \Vert _{\gamma}^{\gamma}+ \varepsilon \gamma (1-a)\mathbb{F}(t) \\ &{}+\varepsilon \biggl\{ \frac{\gamma (1-a)}{2} \biggl(\zeta _{0}- \int _{0}^{t}f( \Lambda )\,d\Lambda \biggr)- \biggl( \zeta _{0}-\frac{1}{2} \int _{0}^{t}f( \Lambda )\,d\varrho \biggr)- \frac{c\mu _{1}}{2B_{0}\kappa} \biggr\} \Vert \nabla v \Vert _{2}^{2} \\ &{}+\varepsilon \zeta _{1} \biggl\{ \frac{\gamma (1-a)}{2}-1 \biggr\} \Vert \nabla v \Vert _{2}^{4}+\varepsilon \biggl\{ \frac{\gamma (1-a)}{2}- \frac{1}{2} \biggr\} (f\circ \nabla v) \\ &{}+\frac{\varepsilon \gamma (1-a)}{2} \int _{\Omega} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}s \bigl\vert \mu _{2}(s) \bigr\vert x^{2}(y,\rho ,s,t)\,ds \,d\rho \,dy. \end{aligned}$$

(3.16)

At this point, we take $a>0$ small enough so that

$$ \varrho _{1}=\frac{\gamma (1-a)}{2}-1>0, $$

and we assume that

$$ \int _{0}^{\infty}f(\Lambda )\,d\Lambda < \frac {\frac{\gamma (1-a)}{2}-1}{\frac{\gamma (1-a)}{2}-\frac{1}{2}}= \frac {2\lambda _{1}}{2\lambda _{1}+1} $$

(3.17)

gives

$$\begin{aligned} \varrho _{2} =& \biggl\{ \biggl(\frac{\gamma (1-a)}{2}-1 \biggr)-\biggl( \int _{0}^{t}f( \Lambda )\,d\Lambda \biggr) \biggl( \frac{\gamma (1-a)}{2}-\frac{1}{2} \biggr) \biggr\} >0, \end{aligned}$$

then we select κ in a way that

$$\begin{aligned}& \varrho _{3} = a-\frac{c\mu _{1}}{4C_{0}\kappa}>0, \\& \varrho _{4} = \varrho _{2}-\frac{c\mu _{1}}{4B_{0}\kappa}>0. \end{aligned}$$

Finally, we set a and κ as fixed values and select ε to be sufficiently small fulfilling

$$ \varrho _{5}=(1-\alpha )-\varepsilon \kappa >0 $$

and

$$ \mathcal{L}(0)>0. $$

This implies that for some $\eta >0$ estimate (3.14) becomes

$$\begin{aligned} \mathcal{L}'(t) \geq &\eta \biggl\{ \mathbb{H}(t)+ \Vert v_{t} \Vert _{2}^{2} + \Vert \nabla v \Vert _{2}^{2}+(f\circ \nabla v)+ \Vert v \Vert _{\gamma}^{ \gamma}+ \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy \\ &{}+ \Vert \nabla v \Vert _{2}^{4}+ \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}s \bigl\vert \mu _{2}(s) \bigr\vert \bigl\Vert (x^{2}(y,\rho ,s,t) \bigr\Vert _{2}^{2}\,ds \,d\rho \biggr\} . \end{aligned}$$

(3.18)

Subsequently, employing the inequalities of Holder and Young, we obtain

$$\begin{aligned} \biggl\vert \int _{\Omega}v v_{t}\,dy \biggr\vert ^{\frac{1}{1-\alpha}} \leq &c \bigl[ \Vert v \Vert _{\gamma}^{\frac{\theta}{1-\alpha}}+ \Vert v_{t} \Vert _{2}^{\frac{\mu}{1-\alpha}} \bigr] , \end{aligned}$$

(3.19)

where $\frac{1}{\mu}+\frac{1}{\theta}=1$. We take $\mu =2(1-\alpha )$ to get

$$ \frac{\theta}{1-\alpha}=\frac{2}{2(1-\alpha )-1}\leq \gamma . $$

Further, for $s = \frac{2}{2(1-\alpha )-1}$, estimate (3.19) gives

$$\begin{aligned} \biggl\vert \int _{\Omega}v v_{t}\,dx \biggr\vert ^{\frac{1}{1-\alpha}} \leq &b \bigl[ \Vert v \Vert _{\gamma}^{s}+ \Vert v_{t} \Vert _{2}^{2} \bigr] . \end{aligned}$$

Then, Lemma 2.5 yields

$$\begin{aligned} \biggl\vert \int _{\Omega}v v_{t}\,dx \biggr\vert ^{\frac{1}{1-\alpha}} \leq &c \bigl[ \Vert v \Vert _{\gamma}^{\gamma}+ \Vert v_{t} \Vert _{2}^{2}+ \Vert \nabla v \Vert _{2}^{2} \bigr]. \end{aligned}$$

(3.20)

Hence,

$$\begin{aligned} \mathcal{L}^{\frac{1}{1-\alpha}}(t) =& \biggl(\mathbb{F}^{1-\alpha}+ \varepsilon \int _{\Omega}vv_{t}\,dy+\frac{\varepsilon \mu _{1}}{2} \Vert v \Vert ^{2}_{2}+\varepsilon \frac{\sigma}{4} \Vert \nabla v \Vert ^{4}_{2} \biggr)^{\frac{1}{1-\alpha}} \\ \leq &c \biggl(\mathbb{F}(t)+ \biggl\vert \int _{\Omega}v v_{t} \,dy \biggr\vert ^{ \frac{1}{1-\alpha}}+ \Vert v \Vert ^{\frac{2}{1-\alpha}}_{2}+ \Vert \nabla v \Vert ^{\frac{4}{1-\alpha}}_{2} \biggr) \\ \leq &c \bigl(\mathbb{F}(t)+ \Vert v \Vert _{\gamma}^{\gamma}+ \Vert v_{t} \Vert _{p+2}^{p+2}+ \Vert \nabla v \Vert _{2}^{2}+ \Vert \nabla v \Vert ^{4}_{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} \bigr) \\ \leq &c \biggl(\mathbb{F}(t)+ \Vert v \Vert _{\gamma}^{\gamma}+ \Vert v_{t} \Vert _{2}^{2}+ \Vert \nabla v \Vert _{2}^{2}+ \Vert \nabla v \Vert ^{4}_{2}+(f \circ \nabla v) \\ &{}+ \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}s \bigl\vert \mu _{2}(s) \bigr\vert \bigl\Vert x^{2} (y,\rho ,s,t) \bigr\Vert _{2}^{2} \biggr) \,ds \,d\rho + \int _{\Omega} \vert v \vert ^{\gamma}\ln \vert v \vert ^{l} \,dy. \end{aligned}$$

(3.21)

Next, (3.18) and (3.21) implies

$$ \mathcal{L}'(t)\geq \Gamma \mathcal{L}^{\frac{1}{1-\alpha}}(t) $$

(3.22)

with $\Gamma > 0$, this relies on η and b only.

By integration of (3.22), we have

$$ \mathcal{L}^{\frac{\alpha}{1-\alpha}}(t)\geq \frac{1}{\mathcal{L}^{\frac{-\alpha}{1-\alpha}}(0)-\Gamma \frac{\alpha}{(1-\alpha )} t}. $$

Hence, $\mathcal{L}(t)$ blows up in time

$$ P\leq P^{*}= \frac{1-\alpha}{\Gamma \alpha \mathcal{L}^{\alpha /(1-\alpha )}(0)}. $$

The proof is completed. □

4 Conclusion

The examination of solutions to the Kirchhoff equation in the context of viscoelastic materials holds paramount significance. In our investigation, a Kirchhoff-type equation featuring nonlinear viscoelastic properties, distinguished by distributed delay, logarithmic nonlinearity, and Balakrishnan–Taylor damping terms, was examined. Subsequent to verifying pertinent hypotheses, the manifestation of solution blow-up was conclusively established.