Abstract
In this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form
There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.
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1 Introduction
We consider the following wave equation with frictional damping, time delay in the velocity, and logarithmic source:
where \(\Omega \subset {\mathbb{R}}^{N} \), \(N \geq 1\), is a bounded domain with smooth boundary ∂Ω. \(\tau >0\) is time delay, α, β, and γ are real numbers that will be specified later. Equation (1.1) is related to a relativistic version of logarithmic quantum mechanics and many branches of physics such as nuclear physics, optics and geophysics [3, 10, 15].
One of the important theories addressing the existence and nonexistence of solutions for problems with source terms is the potential well method, which was devised by Sattinger [29]. Based on the method, the interaction between the damping and the source terms was firstly considered by Levine [16]. Since then, the damped wave equation with polynomial nonlinear source of the form
has been studied extensively on existence, nonexistence, stability, and blow-up of solutions (see [4, 12, 13, 30] and the references therein). Recently, much attention has been paid to the study of nonlinear models of hyperbolic and parabolic equations with logarithmic source nonlinearity [1, 2, 5–8, 17, 18, 22]. For the strongly damped wave equation
Ma and Fang [22] showed the global existence and infinite time blow-up of solutions when \(\gamma =2\), \(a =1\), and \(b =0\). They used a family of potential wells that is related to the logarithmic nonlinearity, which was introduced by Chen et al. [7]. Lian and Xu [18] proved the global existence, energy decay and infinite time blow-up of solutions when \(\gamma =1\), \(a \geq 0\), and \(b > - a \lambda \), where λ is the first eigenvalue of the operator −Δ under homogeneous Dirichlet boundary conditions. In [1], the authors considered the plate equation
They proved the global existence of solutions and showed that the solutions decay exponentially for a suitable initial data. Later, they extended the results to the case of nonlinear damping in the work [2]. There is not much literature for wave equations with time delay and logarithmic nonlinear source. Thus, in this paper, we intend to study such problem; see (1.1)–(1.4). When \(\gamma =0\) in (1.1), Nicaise and Pignotti [24] proved that the energy decays exponentially under the condition \(0< \beta < \alpha \), and then improved the result to the case of time varying delay in [25]. For related work on problems with time delay, we also refer to [9, 14, 27, 31, 32] and the references therein. Inspired by these results, we discuss the solutions for problem (1.1)–(1.4). To the best of our knowledge, there is little work that takes into account wave equations with time delay and logarithmic source. Thus, we prove the local existence of solutions for problem (1.1)–(1.4) via Faedo–Galerkin’s method and the logarithmic Sobolev inequality, and then show the global existence and energy estimates of solutions using the perturbed energy method. Moreover, we establish an infinite time blow-up result by applying the ideas presented in [20, 23, 26] with some necessary modification.
The outline of this paper is as follows. In Sect. 2, we give some notations and material needed for our work. In Sect. 3, we prove the local existence for problem (1.1)–(1.4). In Sect. 4, we provide the global existence and energy decay rates of solutions. Finally, in Sect. 5, we show that the solution occurs with an infinite time blow-up.
2 Preliminaries
We denote the norm of X by \(\| \cdot \|_{X} \) for a Banach space X. We denote the scalar product in \(L^{2} (\Omega )\) by \((\cdot , \cdot )\). For brevity, we denote \(\| \cdot \|_{2}\) by \(\| \cdot \| \). Let \(B_{1}\) be the optimal constant of the embedding inequality
With regard to problem (1.1)–(1.4), we impose the following assumptions:
- \((H_{1})\):
-
The weights of dissipation and delay satisfy
$$ 0 < \vert \beta \vert < \alpha . $$(2.2) - \((H_{2})\):
-
The constant γ in (1.1) satisfies
$$ 0< \gamma < \pi e^{\frac{2(N+1)}{N}} . $$(2.3)
Let us list some lemmas for our work.
Lemma 2.1
(Logarithmic Sobolev inequality [7, 11])
For any \(u \in H^{1}_{0}(\Omega )\) and any positive real number k,
Remark 2.1
Even though the inequality (2.4) holds for all \(k > 0 \), for the computations throughout this work, we take the constant k satisfying
where μ is any real number with
Lemma 2.2
(Logarithmic Gronwall inequality [5])
Let \(c>0\) and \(l \in L^{1}(0,T; {\mathbb{R}}^{+})\). If a function f: \([0,T] \to [1, \infty )\) satisfies
then
For \(v \in H^{1}_{0} (\Omega )\), we define
then
Let
then it satisfies, see e.g. [6, 21, 28],
where \(\mathcal{N}\) is the well-known Nehari manifold, given by
Lemma 2.3
For any \(v \in H^{1}_{0}(\Omega )\) with \(\| v \| \neq 0 \), the functions I and J satisfy
where
Proof
By direct computation, we have, for \(\lambda \geq 0\),
and hence we get the desired result. □
Remark 2.2
For a given \(v \in H^{1}_{0}(\Omega )\), \(J(\lambda v)\) has the absolute maximum value at \(\lambda ^{*}\), that is,
Lemma 2.4
The potential depth d in (2.10) satisfies
Proof
From Lemma 2.1, (2.1), and (2.5), we get
Taking the limit \(k \to \sqrt{ \frac{ \pi }{ \gamma } }^{ -} \), we have
Considering this and (2.12), we have
and hence
Thus, we obtain from (2.13) and (2.9)
By the definition of d given in (2.10), we get the desired result. □
3 Local existence of solutions
In this section we prove the local existence of solutions by applying the ideas in [1, 24]. Using the function
problem (1.1)–(1.4) is rewritten as
Definition 3.1
Let \(T>0\). We say that \((u,y)\) is a local solution of problem (3.2)–(3.6) if it satisfies the following:
and
Theorem 3.1
Assume that \((H_{1})\) and \((H_{2})\) hold. Then, for the initial data \(u_{0} \in H^{1}_{0} (\Omega )\), \(u_{1} \in L^{2}(\Omega )\), \(y_{0} \in L^{2}(\Omega \times (0,1)) \), there exists a local solution \((u,y)\) of problem (3.2)–(3.6).
Proof
Let \(\{ v_{i} \}_{i\in {\mathbb{N}}}\) be orthogonal basis of \(H^{1}_{0} (\Omega )\) which is orthonormal in \(L^{2}(\Omega )\). Defining \(\varphi _{i} (x, 0) =v_{i} (x)\), we can extend \(\varphi _{i} (x, 0)\) by \(\varphi _{i} (x, \eta )\) over \(L^{2}( \Omega \times (0,1)) \). We denote \(V_{n} = \operatorname{span} \{ v_{1}, v_{2}, \ldots , v_{n} \}\) and \(W_{n} = \operatorname{span} \{ \varphi _{1}, \varphi _{2}, \ldots , \varphi _{n} \}\) for \(n \geq 1\). We consider the Faedo–Galerkin approximation solution \((u^{n}, y^{n}) \in V_{n} \times W_{n}\) of the form
solving the approximate system
where
Since problem (3.7)–(3.9) is a normal system of ordinary differential equations, there exists a solution \((u^{n}, y^{n})\) on the interval \([0, t_{n})\), \(t_{n} \in (0,T] \). The extension of this solution to the whole interval \([0,T)\) is a consequence of the estimate below.
Replacing v by \(u^{n}_{t}(t) \) in (3.7) and using the relation
we have
Replacing φ by \(\omega y^{n}(\eta ,t) \) in (3.8), one sees
Collecting (3.10) and (3.11), we get
where
here
By Young’s inequality and the fact \(y^{n}(x,0,t)=u^{n}_{t}(x,t)\), we get
and
where
From this and Lemma 2.1, we observe
Thanks to (2.5), we have
and hence
here and in the sequel \(c_{j}\), \(j=1,2, \ldots \) , denotes a generic positive constant. On the other hand, it is noted that
Applying Cauchy–Schwarz’ inequality and (3.17), we get
By Lemma 2.2, we find
Since the function \(f(s)=s \ln s \) is continuous \((0, \infty )\), \({ \lim_{s \to 0^{+}} f(s) =0 }\), \({ \lim_{s \to + \infty } f(s) = +\infty }\), and f decreases on \((0, e^{-1}) \) and increases on \((e^{-1}, + \infty )\), we have from (3.18) and (3.17)
So, there exists a subsequence of \(\{ ( u^{n} , y^{n} ) \} \), which we still denote \(\{ ( u^{n} , y^{n} ) \} \), such that
By Aubin–Lions’ compactness theorem, we find
and
Since the function \(s \to s \ln |s|^{\gamma }\) is continuous on \(\mathbb{R}\),
Now, we let
Then we have
here we used the fact
From (3.25) and (3.17), we arrive at
where \(B_{2}\) is the best Sobolev imbedding constant of
Thus, we have from (3.26)
By the Lebesgue bounded convergence theorem, (3.24), and (3.27), we infer
Now, we are ready to pass to the limit \(m\to \infty \) in (3.7) and (3.8). The proof of the remainder is standard and can be done as in [1, 19]. □
4 Global existence and energy decay estimate
In this section, we prove the global existence and energy decay rates of solutions to problem (3.2)–(3.6). For this, we define the energy of problem (3.2)–(3.6) as
where ω is the positive constant given in (3.12). It is noted that
By the same arguments as of (3.13), we can deduce
where \(C_{1}\) and \(C_{2}\) are positive constants given in (3.15).
Lemma 4.1
Assume that \((H_{1})\) and \((H_{2})\) hold. If \(E(0) < d \) and \(I(u_{0}) >0 \), then the solution u of problem (1.1)–(1.4) satisfies
where T is the maximal existence time of the solutions.
Proof
Since \(I(u_{0}) >0\) and u is continuous on \([0, T)\), we know that
Let \(t_{0}\) be the maximum of \(t_{1}\) satisfying (4.5). Suppose \(t_{0} < T \), then \(I(u(t_{0})) =0 \), that is,
Thus, we have from (2.11)
But this is contradiction to the following relation:
□
It is noted that \(E(t)\) is a nonincreasing positive function from (4.3) and Lemma 4.1.
Theorem 4.1
Under the conditions of Lemma 4.1, the solution u is global.
Proof
It suffices to show that \(\|u_{t} (t)\|^{2} + \|\nabla u (t) \|^{2}\) is bounded independent of t. From Lemma 4.1, (4.2), and (4.3), we have
Similarly, we see
From Lemma 2.1 and (2.8), we infer
Taking the limit \(k \rightarrow \rho ^{+}\) in this inequality and using (4.7), we get
From Lemma 2.4 and (2.5), we get
Thus, we observe from (4.8) and (4.7) that
This gives
We complete the proof from (4.6) and (4.9). □
In order to establish asymptotic behavior for the global solution, let us define the perturbed energy by
where \(\varepsilon >0 \), \({ \Phi (t) = ( u_{t} (t), u(t) ) }\), and \({ \Xi (t) = \int _{\Omega } \int ^{1}_{0} e^{- \tau \eta } y^{2}(x,\eta ,t) \,d\eta \,dx. } \)
Lemma 4.2
If the conditions of Lemma 4.1hold, there exist positive constants \(C_{3}\) and \(C_{4}\) such that
Proof
Young’s inequality and Lemma 4.1 imply
Taking \(\varepsilon >0\) suitably small, we complete the proof. □
Theorem 4.2
Let \((H_{1})\) and \((H_{2})\) hold. Assume that \(E(0) < E_{1} \) and \(I(u_{0}) >0 \). Then there exist positive constants \(C_{0}\) and \(C_{5} \) such that
Proof
Using (3.2) and Young’s inequality, we have
From (3.3) and the integration by parts, we get
Collecting these and (4.3), we have
Subtracting and adding \(\xi E(t)\) with \(0 < \xi < 2 \varepsilon \), we have
From the logarithmic Sobolev inequality, we obtain
First, we choose \(\varepsilon >0\) small such that
Then, taking \(\xi >0 \) sufficiently small and noting that \(\frac{1}{2} - \frac{\gamma k^{2} }{2 \pi } >0 \) (see (2.5)), we arrive at
Since \(0< E(0) < E_{1}\), there exists \(0 < \mu <1\) such that \(E(0) = \mu E_{1} \). Thus, we have from (4.7)
Thus, we infer from (2.5) that
Substituting this into (4.10), we conclude
Consequently, we complete the proof from Lemma 4.2. □
5 Infinite time blow-up
In this section, inspired by the ideas in [20, 23, 26], we establish a blow-up result for problem (1.1)–(1.4). For this, we first give the following lemma.
Lemma 5.1
Assume that \((H_{1})\) and \((H_{2})\) hold. If \(E(0) < E_{1} \) and \(I(u_{0}) <0 \), then the solution u of problem (1.1)–(1.4) satisfies
and
where T is the maximal existence time of solutions.
Proof
Since \(I(u_{0}) < 0\) and u is continuous on \([0, T)\), we know that
Let \(t_{0}\) be the maximal time satisfying (5.3) and suppose \(t_{0} < T \), then \(I(u(t_{0})) =0 \), that is,
Thus, we have
This is in contradiction to Lemma 2.4. So, (5.1) is proved. From Lemma 2.4, (2.13), and (5.1), we find
Thus, we complete the proof. □
Theorem 5.1
Assume that \((H_{1})\) and \((H_{2})\) hold. Assume that \(E(0) < \zeta E_{1} \), where \(0< \zeta < 1 \), and \(I(u_{0}) <0 \). Then the solution of problem (1.1)–(1.4) blows up at infinity.
Proof
We set
From (4.3), we have
From (5.5), (4.1), and (5.2), we observe
Now, we define
By Young’s inequality and (5.5), we get
Adapting this to (5.7) and using (5.4) and (5.2), we get
First, we fix \(\delta >0\) such that \({ (1 - \zeta ) \frac{\gamma }{2} - |\beta | \delta >0 }\), then choose \(\varepsilon >0\) sufficiently small so that \({ 1 - \frac{\varepsilon | \beta |}{4 \delta C_{2}} > 0 } \). Then we have from (5.5)
On the other hand, we can easily see that
Let us take \(\varepsilon >0\) sufficiently small again to get
Then we obtain from (5.9) and (5.11)
From (5.9) and (5.10), we observe
and hence
Thus, \(G(t)\) blows up at infinity. □
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The author is grateful to the anonymous referees for the careful reading and their important comments to improve this paper.
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This research 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. Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source. Adv Differ Equ 2020, 631 (2020). https://doi.org/10.1186/s13662-020-03037-6
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DOI: https://doi.org/10.1186/s13662-020-03037-6