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General decay and blow-up of solutions for a nonlinear wave equation with memory and fractional boundary damping terms
Boundary Value Problems volume 2020, Article number: 172 (2020)
Abstract
The paper studies the global existence and general decay of solutions using Lyapunov functional for a nonlinear wave equation, taking into account the fractional derivative boundary condition and memory term. In addition, we establish the blow-up of solutions with nonpositive initial energy.
1 Introduction
Extraordinary differential equations, also known as fractional differential equations, are a generalization of differential equations through fractional calculus. Much attention has been accorded to fractional partial differential equations during the past two decades due to the many chemical engineering, biological, ecological, and electromagnetism phenomena that are modeled by initial boundary value problems with fractional boundary conditions. See Tarasov [16], Magin [15].
In this work we consider the nonlinear wave equation
where Ω is a bounded domain in \(\mathbb{R} ^{n}\), \(n\geq 1\) with a smooth boundary ∂Ω of class \(C^{2}\) and ν is the unit outward normal to \(\partial \Omega =\Gamma _{0}\cup \Gamma _{1}\), where \(\Gamma _{0}\) and \(\Gamma _{1}\) are closed subsets of ∂Ω with \(\Gamma _{0}\cap \Gamma _{1}=\emptyset \).
\(a,b>0\), \(p>2\), and \(\partial _{t}^{\alpha ,\eta }\) with \(0<\alpha <1\) is the Caputo’s generalized fractional derivative (see [11] and [7]) defined by
where Γ is the usual Euler gamma function. It can also be expressed by
where \(I^{\alpha ,\eta }\) is the exponential fractional integro-differential operator given by
In the context of boundary dissipations of fractional order problems, the main research focus is on asymptotic stability of solutions starting by writing the equations as an augmented system. Then, various techniques are used such as LaSalle’s invariance principle and the multiplier method mixed with frequency domain (see [1–16], and [18]).
Dai and Zhang [7] replaced \(\int _{0}^{t}K(x,t-s)u_{s}(x,s)\,ds\) with \(\partial _{t}^{\alpha }u(x,t)\) and \(h(x,t)\) with \(|u|^{m-1}u(x,t)\) and managed to prove exponential growth for the same problem.
Note that the nonlinear wave equation with boundary fractional damping case was first considered by authors in [4], where they used the augmented system to prove the exponential stability and blow-up of solutions in finite time.
Motivated by our recent work in [4] and based on the construction of a Lyapunov function, we prove in this paper under suitable conditions on the initial data the stability of a wave equation with fractional damping and memory term. This technique of proof was recently used by [9] and [4] to study the exponential decay of a system of nonlocal singular viscoelastic equations.
Here we also consider three different cases on the sign of the initial energy as recently examined by Zarai et al. [17], where they studied the blow-up of a system of nonlocal singular viscoelastic equations.
The organization of our paper is as follows. We start in Sect. 2 by giving some lemmas and notations in order to reformulate our problem (1.1) into an augmented system. In the following section, we use the potential well theory to prove the global existence result. Then, the general decay result is given in Sect. 4. In Sect. 5, following a direct approach, we prove blow-up of solutions.
2 Preliminaries
Let us introduce some notations, assumptions, and lemmas that are effective for proving our results.
Assume that the relaxation function g satisfies
\(( G_{1} ) \) \(g:\mathbb{R} _{+}\rightarrow \mathbb{R} _{+}\) is a nonincreasing differentiable function with
\(( G_{2} ) \) There exists a constant \(\xi >0\) such that
We denote
and
Lemma 1
(Sobolev–Poincaré inequality)
If either \(1\leq q\leq \frac{N+2}{N-2}\) \(( N\geq 3 ) \) or \(1\leq q\leq +\infty \) \(( N=2 ) \), then there exists \(C_{\ast }>0\) such that
Lemma 2
(Trace–Sobolev embedding)
For all p such that
we have
We denote by \(B_{q}\) the embedding constant, i.e.,
Lemma 3
([17], p. 5, Lemma 2 or [3], p. 1406, Lemma 4.1)
Consider a nonnegative function \(B(t)\in C^{2}(0,\infty )\) satisfying
where \(\delta >0\).
If
then
where \(l_{0} \in \mathbb{R}\), \(r_{2}\) represents the smallest root of the equation
i.e., \(r_{2}=2(\delta +1)-2\sqrt{(\delta +1)\delta }\).
Lemma 4
([17], p. 5, Lemma 3 or [3], p. 1406, Lemma 4.2)
Let \(J (t ) \) be a nonincreasing function on \([ t_{0},\infty ) \) verifying the differential inequality
where \(\alpha >0\), \(b\in \mathbb{R} \), then there exists \(T^{\ast } >0\) such that
with the following upper bound cases for \(T^{\ast }\):
\(\mathbf{(i)}\) When \(b<0\) and \(J(t_{0})<\min \{ 1,\sqrt{\alpha /(-b)} \} \),
\(\mathbf{(ii)}\) When \(b=0\),
\(\mathbf{(iii)}\) When \(b>0\),
or
where
Definition 1
We say that u is a blow-up solution of (1.1) at finite time \(T^{\ast }\) if
Theorem 1
([12], Theorem 1)
Consider the constant
and the function μ given by
Then we can obtain
which is a relation between U the “input” of the system
and the “output” O given by
Now, using (1.2) and Theorem 1, the augmented system related to our system (1.1) may be given by
where \(b_{1}=b\varrho \).
Lemma 5
([2], p. 3, Lemma 2.1)
For all \(\lambda \in D_{\eta }= \{ \lambda \in \mathbf{\mathbb{C}}:\Im m\lambda \neq 0 \} \cup \{ \lambda \in \mathbf{\mathbb{C}}:\Re e\lambda +\eta >0 \} \), we have
Theorem 2
(Local existence and uniqueness)
Assume that (2.4) holds. Then, for all \((u_{0},u_{1},\phi _{0})\in H_{\Gamma _{0}}^{1}(\Omega )\times L^{2}( \Omega )\times L^{2}(-\infty ,+\infty )\), there exists some T small enough such that problem (2.20) admits a unique solution
3 Global existence
Before proving the global existence for problem (2.20), let us introduce the functionals
and
The energy functional E associated with system (2.20) is given as follows:
Lemma 6
If \((u,\phi )\) is a regular solution to (2.20), then the energy functional given in (3.1) verifies
Proof
Multiplying by \(u_{t}\) in the first equation from (2.20), using integration by parts over Ω, we get
Therefore
Multiplying by \(b_{1}\phi \) in the second equation from (2.20) and integrating over \(\Gamma _{0}\times (-\infty ,+\infty )\), we get
From (3.1), (3.3), and (3.4) we obtain
□
Lemma 7
Assuming that (2.4) holds and that for all \((u_{0},u_{1},\phi _{0})\in H_{\Gamma _{0}}^{1}(\Omega )\times L^{2}( \Omega )\times L^{2}(-\infty ,+\infty )\) verifying
Then \(u(t)\in \aleph \), \(\forall t\in {[} 0,T]\).
Proof
As \(I(u_{0})>0\), there exists \(T^{\ast }\leq T\) such that
This leads to
Using the Poincare inequality, (3.1), (2.3), (3.5), and (3.6), we obtain
Thus
Consequently, \(u\in H\), \(\forall t\in {[} 0,T^{\ast })\).
Repeating the procedure, \(T^{\ast }\) can be extended to T, and that makes the proof of our global existence result within reach. □
Theorem 3
Assume that (2.4) holds. Then for all
verifying (3.5), the solution of system (2.20) is global and bounded.
Proof
From (3.2), we get
Or \(I(t)>0\), therefrom
where \(C_{1}=\max \{\frac{2}{b_{1}},\frac{2p}{p-2},2\}\). □
4 Decay of solutions
To proceed for the energy decay result, we construct an appropriate Lyapunov functional as follows:
where
and \(\epsilon _{1}\), \(\epsilon _{2}\) are positive constants.
Lemma 8
If \((u,\phi )\) is a regular solution of problem (2.20), then the following equality holds:
Proof
From the second equation of (2.20), we have
Integrating (4.2) over \([0, t ]\) and using equations 3 and 6 from system (2.20), we get
hence,
Multiplying by ϕ followed by integration over \(\Gamma _{0}\times (-\infty ,+\infty )\) leads to
□
Lemma 9
For any \((u,\phi )\) solution of problem (2.20), we have
where \(\alpha _{1}\), \(\alpha _{2}\) are positive constants.
Proof
From (4.3), we get
Thus
Multiplying by \(\xi ^{2}+\eta \) in (4.7) followed by integration over \(\Gamma _{0}\times (-\infty ,+\infty )\) leads to
Using Young’s inequality in order to have an estimation of the last term in (4.8), we get for any \(\delta >0\)
Combining (4.8) and (4.9), we obtain
Since \(\frac{1}{\xi ^{2}+\eta }\leq \frac{1}{\eta }\), then
Applying Lammas 2 and 5, we get
By Poincare-type inequality and Young’s inequality, we obtain
Adding (4.13) to (4.12), we get
Therefore, by the energy definition given in (3.1), for all \(N>0\), we have
From (3.7) and (4.15), we finally get
where N and \(\epsilon _{1}\) are chosen as follows:
Then we conclude from (4.16)
where
and
□
Now, we prove the exponential decay of global solution.
Theorem 4
If (2.4) and (3.5) hold, then there exist k and K, positive constants such that the global solution of (2.20) verifies
Proof
By differentiation in (4.1), we get
Combining with (2.20) to obtain
An application of Lemma 8 leads to
Using Poincare-type inequality and Young’s inequality on the last term of (4.20), we get, for all \(\delta ^{\prime }>0\),
From (4.20), (4.21), and (3.2), we obtain
We use (3.7) to get
On the other hand, from (3.5)
For a small enough \(\delta ^{\prime }\), we may have
Then choose \(d>0\) depending only on \(\delta ^{\prime }\) such that
Equivalently, for all positive constant M, we have
For \(\epsilon _{1}\) and \(M<\min \{2,2\,d\}\) chosen such that
We obtain from (4.25)
as a result of (4.5). Now, a simple integration of (4.26) yields
where \(k=\frac{\epsilon _{2}M}{\alpha _{2}}\). Another use of (4.5) provides (4.17). □
5 Blow-up
In the current section, we follow the same approach given in [11] to prove the blow-up of solution of problem (2.20).
Remark 1
By integration of (3.2) over \((0,t)\), we have
Now, let us define \(F(t)\):
where
Lemma 10
Assume that \(\| \nabla u\| _{2}^{2}\) is bounded on \([0,T)\), then
More precisely
where
Proof
Using (2.18), we obtain
Hölder’s inequality yields
On the other hand,
From (5.6) in (5.7), we obtain
Applying Lemma 2 leads to
Since \(z\in (0,s)\), we choose \(\exists C_{2}\geq 0\) such that \(s-z\geq \frac{C_{2}}{2}\) to term (5.9) into
Multiplication by \(\xi ^{2}+\eta \) followed by integration over \((0,t)\times (-\infty ,+\infty )\) yields
Then
Applying a special integral (Euler gamma function), we obtain
□
Lemma 11
Suppose \(p>2\), then
Proof
We differentiate with respect to t in (5.2), then we get
Using divergence theorem and (2.20), we obtain
By definition of energy functional in (3.1) and relation (5.1), we give the following evaluation of the third term of (5.16):
We can also estimate the last term of (5.16) using Lemma 8:
From (5.17), (5.18), and (5.16), we get
Taking \(p>2\), we obtain the needed estimation
□
Lemma 12
Suppose that \(p>2\) and that either one of the next assumptions is verified:
(i) \(E(0)<0\);
(ii) \(E(0)=0\), and
(iii) \(E(0)>0\), and
where
and
Then \(F^{\prime }(t)>a\| u_{0}\| _{2}^{2}\) for \(t>t_{0}\), where
where \(t_{0}=t^{\ast }\) in case (i), and \(t_{0}=0\) in cases (ii) and (iii).
Proof
(i) Case of \(E(0)<0\).
From (5.14), we have
which clearly leads to
Then
where \(t^{\ast }\) as given in (5.23).
(ii) Case \(E(0)=0\).
Using (5.14) we get
Thus
Then, by (5.20),
(iii) Case \(E(0)>0\).
The proof of this case consists of getting to a differential inequality: \(B^{\prime \prime }(t)-pB^{\prime }(t)+pB(t)\geq 0\) pursued by a use of Lemma 3. Indeed, from (5.15) we have
Or, the last term in (5.24) can be estimated using Young’s inequality
On the other hand,
By Young’s inequality, we get
Now, we remake (5.24) using (5.25) and (5.27):
From the definition of F in (5.2), inequality (5.28) also becomes
Thus, by (5.14), we get
Hence
where
Posing
leads to
By Lemma 3 and for \(p=\delta +1\), we conclude that if
then
□
Theorem 5
Suppose that \(p>2\) and that either one of the next assumptions is verified:
(i) \(E(0)<0\);
(ii) \(E(0)=0\) and (5.20) holds;
(iii) \(0< E(0)< \frac{(2p-4) ( F^{\prime }(t_{0})-a\| u_{0}\| _{2}^{2} ) ^{2}J(t_{0})^{\frac{1}{\gamma _{1}}}}{16p}\) and (5.21) holds.
Then, in the sense of Definition 1, the solution \((u,\phi )\) blows up at finite time \(T^{\ast }\).
For case (i):
Moreover, if \(J(t_{0})<\min \{ 1,\sqrt{\frac{\sigma }{-b}} \} \), we get
For case (ii), we get either
or
For case (iii):
or else
where \(\gamma _{1}=\frac{p-4}{4}\), \(c=(\frac{b}{\sigma })^{\frac{\gamma _{1}}{2+\gamma _{1}}}\), \(J(t)\), b and σ are as in (5.40) and (5.54) respectively.
Note that \(t_{0} =0\) in cases (ii) and (iii). For case (i), we have as in (5.23): \(t_{0}=t^{*}\).
Proof
Consider
We differentiate on \(J(t)\) to get
and again
where
Using (5.14), we obtain
Consequently,
Or, from (5.15) and the fact that \(\| u\| _{2}^{2}-\| u_{0}\| _{2}^{2}=2\int _{0}^{t}\int _{\Omega }u_{s}u\,dx \,ds\), we attain
Going back to (5.43) with (5.44) and (5.45) in hand, we get
For the sake of simplicity, we introduce the following notations:
Therefore
Note that, \(\forall w\in R\) and \(\forall t>0\),
Hence
It is clear that
and
Then, from (5.47) and (5.50), we obtain
Or, by Lemma [6], \(J^{{\prime }}(t)<0\), where \(t\geq t_{0}\).
Multiplication by \(J^{{\prime }}(t)\) in (5.52), followed by integration from \(t_{0}\) to t, leads to
where
Note that \(\sigma >0\) is equivalent to \(E(0)< \frac{(2p-4) ( F^{\prime }(t_{0})-a\| u_{0}\| _{2}^{2} ) ^{2}J(t_{0})^{\frac{1}{\gamma _{1}}}}{16p}\), which by Lemma 4 ensures the existence of a finite time \(T^{\ast }>0\) such that
That involves
i.e.,
So, there exists T such that \(t_{0}< T\leq T^{*}\) and \(\| \nabla u\| _{2}^{2}\longrightarrow +\infty \) as \(t\longrightarrow T^{-}\).
Indeed, if it is not the case, then \(\| \nabla u\| _{2}^{2}\) remained bounded on \([t_{0},T^{\ast })\), which by Lemma 10 leads to
contradicting (5.56). □
6 Conclusion
Much attention has been accorded to fractional partial differential equations during the past two decades due to the many chemical engineering, biological, ecological, and electromagnetism phenomena that are modeled by initial boundary value problems with fractional boundary conditions. In the context of boundary dissipations of fractional order problems, the main research focus is on asymptotic stability of solutions starting by writing the equations as an augmented system. Then, various techniques are used such as LaSalle’s invariance principle and the multiplier method mixed with frequency domain. We prove in this paper under suitable conditions on the initial data the stability of a wave equation with fractional damping and memory term. This technique of proof was recently used by [4] to study the exponential decay of a system of nonlocal singular viscoelastic equations. Here we also considered three different cases on the sign of the initial energy as recently examined by Zarai et al. [17], where they studied the blow-up of a system of nonlocal singular viscoelastic equations.
In the next work, we will try to extend the same study of this paper to a general source term case.
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Acknowledgements
The authors are grateful to the anonymous referees for the careful reading and their important observations/suggestions for the sake of improving this paper. The first author (Pr. Salah Boulaaras) would like to thank all the professors of the mathematics department at the University of Annaba in Algeria, especially his Professors/Scientists Pr. Mohamed Haiour, Pr. Ahmed-Salah Chibi, and Pr. Azzedine Benchettah for the important content of masters and PhD courses in pure and applied mathematics which he received during his studies. Moreover, he thanks them for the additional help they provided to him during office hours in their office about the few concepts/difficulties he had encountered, and he appreciates their talent and dedication for their postgraduate students currently and previously.
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On the occasion of the 44th birthday of the first author’s brother, Professor Djemai Mahmoud Mouha Boulaaras.
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Boulaaras, S., Kamache, F., Bouizem, Y. et al. General decay and blow-up of solutions for a nonlinear wave equation with memory and fractional boundary damping terms. Bound Value Probl 2020, 172 (2020). https://doi.org/10.1186/s13661-020-01470-w
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DOI: https://doi.org/10.1186/s13661-020-01470-w