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Blowup for nonlinearly damped viscoelastic equations with logarithmic source and delay terms
Advances in Difference Equations volume 2021, Article number: 316 (2021)
Abstract
In this work, we investigate blowup phenomena for nonlinearly damped viscoelastic equations with logarithmic source effect and time delay in the velocity. Owing to the nonlinear damping term instead of strong or linear dissipation, we cannot apply the concavity method introduced by Levine. Thus, utilizing the energy method, we show that the solutions with not only non-positive initial energy but also some positive initial energy blow up at a finite point in time.
1 Introduction
We discuss the viscoelastic wave equation with nonlinear damping, logarithmic source, and delay terms
here \(\Omega \subset {\mathbb{R}}^{n} \) is a bounded domain with smooth boundary ∂Ω, \(k * \Delta u = \int ^{t}_{0} k (t-s) \Delta u(s) \,ds\), the kernel function \(k : [0,\infty ) \to (0, \infty )\) is a \(C^{1}\)-function with
\(\tau >0\) is time delay, the coefficients \(c_{1} >0\) and \(c_{2} \in {\mathbb{R}} \) satisfy \(0 < | c_{2} | < c_{1} \), and the exponents \(q \geq 2\) and \(p >2 \) are specified later.
Many researchers have studied parabolic or hyperbolic equations with logarithmic nonlinearity [2, 4, 7, 13, 15]. For the physical application of this nonlinearity, we refer to [1, 6]. In [4], the authors discussed a strongly damped equation,
with Dirichlet boundary condition. They showed that the solutions with subcritical and critical initial energy blow up in a finite point under suitable conditions. Moreover, they estimated bounds of the blowup time. The authors of [7] proved similar results to those of [4] for the equation with memory. Most work dealing with wave equations with logarithmic nonlinearity is associated with a strongly or linearly damped mechanism, and blowup results are investigated by virtue of the potential well method and Levine’s concavity technique [12].
On the other hand, time delay effect arises in many natural phenomena depending not only on the present state but also on some past occurrences. Thus, partial differential equations with time delay have become an active area of research in resent years. For the physical application of the time delay, we refer to [3, 18]. Recently, Kafini and Messaoudi [8] considered the wave equation with linear damping and delay terms
with Dirichlet boundary condition. They established a blowup result of the solution with negative initial energy by adapting the energy method. While there are many studies on the existence and asymptotic stability of the solutions of wave equations with delay, there are relatively few studies on blowup. We refer to [5, 10, 16, 17, 19] and [9] for stability and blowup of equations with delay, respectively. Motivated by this pioneering work [8], in this article, we study blowup phenomena for the nonlinearly damped viscoelastic wave equation (1.1)–(1.4) with logarithmic source effect and time delay in the velocity. Due to the presence of nonlinear dissipation instead of strong or linear damping terms, we cannot apply the concavity method. Thus, by applying the energy method, we establish a blowup result of solutions with not only non-positive initial energy but also some positive initial energy. And, it is worth to mention that there are few works dealing with viscoelastic wave equations with nonlinear damping and logarithmic source terms.
Here is the outline of this paper. In Sect. 2, we present notations, hypotheses, and auxiliary functions and lemmas. In Sect. 3, we establish a blowup criterion of solutions with not only non-positive initial energy but also some positive initial energy.
2 Preliminaries
Throughout this article, \(( \cdot , \cdot ) \) denotes the scalar product in Hilbert space \(L^{2}(\Omega ) \). \(\| \cdot \|_{r} \) represents the norm in the space \(L^{r}(\Omega ) \). Moreover, \(\| \cdot \|_{Y} \) denotes the norm of a normed space Y. \(C >0 \) represents a generic constant. If there is no ambiguity, we omit the variables t and x.
We let the function y be as in [16]
Then problem (1.1)–(1.4) reads
By the arguments of [7, 9], we can state the well-posedness.
Theorem 2.1
Let \((u_{0}, u_{1}, y_{0} ) \in H_{0}^{1}(\Omega ) \times L^{2}(\Omega ) \times L^{q} ( \Omega \times (0,1))\), \(q \geq 2\), and
Then problem (2.1)–(2.5) admits a unique local solution \((u, y) \) with \(u \in C(0,T; H_{0}^{1}(\Omega ) ) \cap C^{1} ( 0,T; L^{2}(\Omega )) \) and \(y \in L^{\infty } ( 0,T; L^{q} ( \Omega \times ( 0,1) ) )\).
Our goal is to find a blowup result to problem (2.1)–(2.5). For this, we will often use the embedding
and Young’s inequality
where
Also, we need the lemmas below, which are proved by Kafini and Messaoudi [8], to estimate logarithmic nonlinearity.
Lemma 2.1
For \(\phi \in L^{p}(\Omega ) \cap H_{0}^{1}(\Omega ) \), we have
Lemma 2.2
For \(\phi \in L^{p+1}(\Omega ) \cap H_{0}^{1}(\Omega ) \) with \(\int _{\Omega } | \phi |^{p} \ln |\phi | \,dx \geq 0 \), we have
Lemma 2.3
For \(\phi \in L^{p}(\Omega ) \cap H_{0}^{1}(\Omega ) \) with \(\int _{\Omega } | \phi |^{p} \ln |\phi | \,dx \geq 0 \), we have
To establish our desired blowup result, we impose the following assumptions:
- \((H_{n}) \):
-
Let \(1 \leq n \leq 5\).
- \((H_{p}) \):
-
Let p satisfy
$$\begin{aligned} \textstyle\begin{cases} 2 < p < \infty & \text{if } n =1,2 ; \\ 2 < p < \min \{ \frac{ 2 ( n -1 )}{ n -2 } , \frac{ n + 2 }{ n -2 } \} & \text{if } n =3, 4, 5 . \end{cases}\displaystyle \end{aligned}$$ - \((H_{q}) \):
-
Let q verify
$$ \max \biggl\{ 2, \frac{p^{2} + 2p}{ p^{2} -2p + 4 } \biggr\} < q < p . $$(2.7) - \((H_{k}) \):
-
Let k satisfy
$$ \int _{0}^{\infty } k(s) \,ds < \frac{ p( 1 - \lambda ) -2 }{ p( 1 - \lambda ) -2 + \frac{1}{2 \eta }} , $$(2.8)where
$$ 0 < \lambda < \frac{ p - 2 }{p}, \qquad 0 < \eta < \frac{ p (1- \lambda ) }{2}. $$(2.9)
From \((H_{p}) \), there exists \(\mu >0 \) satisfying
This implies
Put \(D_{0} \) be the embedding constant with
We let \({ D = \frac{ D_{0} }{ \sqrt{ k_{l} } } } \) and define a continuously differentiable function K as
and put
Lemma 2.4
For \(p >2 \) and \(\mu >0 \), the function K satisfies
-
(i)
\(K (0) =0 \),
-
(ii)
\({ \lim_{\xi \to \infty } K (\xi ) = - \infty }\),
-
(iii)
\(K' ( \xi ) >0 \) on \((0, \xi _{K} ) \), \(K' (\xi ) < 0 \) on \(( \xi _{K} , \infty ) \),
-
(iv)
K has the maximum value \(K_{\max }\) at \(\xi _{K} \).
Proof
The results (i) and (ii) are clear. Since
we have \(K ' (\xi _{K}) =0 \), \(K' ( \xi ) >0 \) on \((0, \xi _{K} ) \), and \(K' (\xi ) < 0 \) on \(( \xi _{K} , \infty ) \). Thus, K has the maximum value
 □
We also need the following auxiliary result in the proof of our main theorem.
Lemma 2.5
For \(p >2 \), \(\mu >0 \), and \(0 < \lambda < \frac{p-2}{p}\), the \(\xi _{K}\) and \(K_{max}\) verify
Proof
First, we claim
Indeed, if \(p + \mu \leq 4 \), it is clear that \(A > 0 \). If \(p + \mu > 4 \),
This means
So, we also have \(A >0 \) if \(p + \mu > 4 \). The result (2.18) implies
which gives
 □
3 Blowup results
In this part, we search a blowup result of the solution to (2.1)–(2.5) inspired by the ideas in [8, 14].
We define the energy to problem (2.1)–(2.5) by
where
and
Lemma 3.1
Under the conditions of Theorem 2.1, the equation
is fulfilled for some \(\gamma _{1} >0 \).
Proof
Taking the scalar product (2.1) by \(u_{t}\) in \(L^{2}(\Omega )\), we get
Using the estimate
and
we get
Using (2.6) with \(\frac{ q -1}{ q } + \frac{1}{ q } =1 \), we have
From (2.2), we find
From (3.4), (3.5), and (3.6), one sees
Letting
we obtain (3.3) from (3.2). □
Lemma 3.2
Let \(( u , y ) \) be the solution of (2.1)–(2.5). If the initial datum \((u_{0}, y_{0})\) satisfies
there exists \(\xi _{1} > \xi _{K} \) such that
Proof
Set
Using (3.1), (2.11), (2.12), and the relation (Lemma 2.1, [11])
we derive
Since \(E(0) < K_{\max }\), there exists \(\xi _{1} > \xi _{K} \) with \(K (\xi _{1}) = E(0) \). From (3.9), we have
Since K is decreasing on \((\xi _{K}, \infty ) \), (3.10) gives
To show (3.8), we use a contradiction. Suppose there exists \(t_{1} >0\) such that \(\sqrt{ k_{l} } \| \nabla u (t_{1}) \|_{2} < \xi _{1} \). The continuity of u corresponding to t gives the existence of \(t_{0} > 0\) with
From this and (3.9), we get
but this contradicts (3.3). □
Now, we are ready to state our main theorem.
Definition 3.1
We say that the solution \((u, y)\) of problem (2.1)–(2.5) blows up in a finite time if there exists a time \(T^{*} \), \(0 < T^{*} < \infty \), such that
Theorem 3.1
Let \((H_{n})\), \((H_{p})\), \((H_{q}) \), \((H_{k})\), and the assumptions of Lemma 3.2hold. Furthermore, we assume
Then the solution \((u, y) \) to problem (2.1)–(2.5) blows up after finite time.
Proof
Let EÌ… with
From this and (2.17), we see
Define
then
From this, (3.1), (3.12), Lemma 3.2, and the definition of \(\xi _{K}\) and \(K_{\max } \), we obtain
which also ensures
Now, we put
where \(\epsilon >0 \) and
Using (2.6) with \(\frac{ q - 1}{q } + \frac{1}{q} =1 \), we have
and
for any \(\chi >0 \). From these and the relation \(0 < |c_{2}| < c_{1} \), we get
Taking \(\chi = ( \theta R^{-\beta }(t) )^{- \frac{q-1}{q} } \), \(\theta >0\), and applying (3.14), we derive
Using (3.16), (3.15), Lemma 2.3, and (2.6) with \(\frac{q}{p} + \frac{ p - q }{p} =1 \), we find
From (3.18), we see \(0 < \beta \leq \frac{ p - q }{ p ( q -1 )} \), which gives
Thanks to \((H_{p}) \), we also note the solution u to (2.1)–(2.5) belongs to \(L^{p+1}(\Omega )\). So, we can apply Lemma 2.2 to get
Similarly, from (3.18), we see \(\frac{ 2( p - q ) }{ p^{2} ( q - 1 )} \leq \beta \leq \frac{ p - q }{ p ( q - 1)} \), which implies
So, we have
Inserting (3.22) and (3.23) to (3.21), we obtain
From this and (3.20),
Applying this and the estimate
to (3.19), we get
Subtracting and adding the term \(\epsilon \lambda \int _{\Omega } | u |^{p} \ln | u | \,dx \), where λ is given in (2.9), and using (3.1) and (3.13), we get
By \((H_{k})\), we note
and
Firstly, we take \(\theta >0\) appropriately large to guarantee
and
Next, we claim
Indeed, it is seen from Lemma 3.2 that
From (3.11), we have
Thus, we can fix \(\theta >0\) suitably large again to get
Finally, we pick \(\epsilon >0\) suitably small to have
and
Therefore, from (3.25) we arrive at
Next, from (3.17), we know
Using (2.6) with \(\frac{1}{ 2(1-\beta )} + \frac{1 - 2 \beta }{ 2(1-\beta )} =1 \), noting \(2 \leq \frac{2}{ 1- 2 \beta } \leq p \) from (3.18), and applying Lemma 2.1, we deduce
Combining (3.29), (3.30) and noting (3.16), we derive
From this and (3.29), we conclude
which shows that the solution u blows up after finite time \(T^{*} \leq \frac{1 - \beta }{ C \beta L^{ \frac{\beta }{1- \beta }} (0) } \). □
4 Conclusion
In this paper, we considered a viscoelastic wave equation with nonlinear damping and time delay terms and logarithmic source effect. Under the conditions \(( H_{n}) \), \(( H_{p} ) \), \(( H_{q} )\), and \((H_{k}) \), we showed that the solutions with not only a non-positive initial energy but also some positive initial energy blow up after a finite time by utilizing the energy method.
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Acknowledgements
The author is grateful to the anonymous referees for their careful reading and important comments.
Funding
This work was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2020R1I1A3066250).
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Park, SH. Blowup for nonlinearly damped viscoelastic equations with logarithmic source and delay terms. Adv Differ Equ 2021, 316 (2021). https://doi.org/10.1186/s13662-021-03469-8
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DOI: https://doi.org/10.1186/s13662-021-03469-8