Local existence and lower bound of blow-up time to a Cauchy problem of a coupled nonlinear wave equations

1.   Introduction

In this work, we consider the following Cauchy problem of a coupled linearly-damped wave equations

where $\lambda _{1}, \lambda _{2}$ are two positive constants. The functions $u_{0},$ $u_{1}, v_{0},$ $v_{1}$ are the initial data to be specified later.

A single equation of problem (1.1), in a bounded domain $\Omega \subset$ $\mathbb{R}^{n}$ $(n\geq 1),$ has been extensively studied and many results concerning global existence and nonexistence have been proved. For instance, for the equation

$m, \gamma \geq 0,$ it is well known that, for $a = 0$, the source term $bu|u|^{\gamma },$ ($\gamma > 0$) causes finite time blow up of solutions with negative initial energy (see ^[1]). The interaction between the damping and the source terms was first considered by Levine ^[2,3] in the linear damping case ($m = 0$). He showed that solutions with negative initial energy blow up in finite time. Georgiev and Todorova ^[4] extended Levine's result to the nonlinear damping case ($m > 0$). In their work, the authors introduced a different method and showed that solutions with negative energy continue to exist globally 'in time' if $m\geq \gamma$ and blow up in finite time if $\gamma > m$ and the initial energy is sufficiently negative. This last blow-up result has been extended to solutions with negative initial energy by Messaoudi ^[5] and others. For results of same nature, we refer the reader to Levine and Serrin ^[6], Vitillaro ^[7], Messaoudi and Said-Houari ^[8] and Messaoudi ^[9,10].

For problem (1.2) in $\mathbb{R}^{n}$, we mention, among others, the work of Levine Serrin and Park ^[11]. Where the authors established a global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation. Todorova ^[12,13], showed that restriction in the case of compactly supported initial data is crucial for the blow-up result and it is not essential for the global existence. The local existence theorem could be proved without the requirement for the compact support of the data. Messaoudi ^[14], consider the Cauchy problem for the nonlinearly damped wave equation with nonlinear source

where $a, b > 0\ $and $m, \gamma > 2$. He proved that given any time $T > 0$, there exist always initial data with sufficiently negative initial energy, for which the solution blows up in time $\leq T.$ This result improves an earlier one by Todorova ^[12]. Zhou ^[15], considered the Cauchy problem for a nonlinear wave equation with linear damping and source terms. He proved that the solution blows up in finite time even for vanishing initial energy if the initial datum $u_{0},$ $u_{1}$ satisfies $\int_{\mathbb{R} ^{n}}u_{0}u_{1}dx\geq 0.$

In ^[16], Kafini and Messaoudi considered the Cauchy problem

with negative initial energy and

they proved a finite-time blow-up result. In ^[17], the same authors proved a blow-up result to a coupled system

Systems in a bounded domains of $\mathbb{R}^{n}$ have been extensively studied by many authors. Messaoudi and Houari ^[18], established a global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. Agre and Rammaha ^[19] studied the following system

in $\Omega \times (0, T)$ with initial and boundary conditions of Dirichlet type. They proved several results concerning local and global existence of a weak solution and showed that any weak solution with negative initial energy blows up in finite time. For more results on different systems, we refer to Messaoudi et al. ^[20], Santos ^[21], Cavalcanti et al. ^[22] and Wu ^[23].

Looking at our system in (1.1) and the system in (1.3), they differ by the type of damping only. But the proof is completely different due to nature of the viscoelastic damping. Although the damping in (1.4) is nonlinear, but it is proved in a bounded domain $\Omega \subset \mathbb{R}^{n}$ and for $n = 1, 2, 3$ only. Also the method is different.

Our aim is to study (1.1) and establish the local existence of solutions, the blow-up result in a finite time using the concavity method introduced by Levine in ^[2]. In addition, we obtain the lower bound of the blow-up time. To achieve this goal some conditions have to be imposed on the source functions $f_{1}$ and $f_{2}$ and the initial data as well. The paper is organized as follows. In section 2, we present conditions needed for our results and the proof of the local existence. In section 3, we present the statement and proof of the main result.

2.   Local existence

We start with the following theorem and assumptions.

Theorem 2.1. ^[24] (Sobolev, Gagliardo, Nirenberg)

Suppose that $1\leq p < n.$ If

with

Moreover there exists a constant C = C(N, p) such that

Corollary 2.2.^[24] Suppose that $1\leq p < n.$ Then

with continuous injection.

(G1) There exists a function $I(u, v)\geq 0$ such that

(G2) There exists a constant $\rho > 2$ such that

(G3)There exists a constant $d > 0$ such that

where

(G4)Further assume that $f_{1}, f_{2}\in C^{1}(\mathbb{R}^{n})$ such that

and $\beta$ $= \max \left\{ \beta _{i}, \ i = 1, 2, 3, 4\right\} > 1.$

Note that condition (G3) and (G4) are sufficient to establish the existence of a local solution of (1.1) in an interval $(0, T]$ (see ^{[20,25,26,27]}) and not needed for the blow-up result.

An example of a function that satisfying the other conditions (G1) and (G2) is

where we have

To start the proof, we set $\phi = u_{t}$, $\psi = v_{t}$ and denote by

Therefore (1.1) can be rewritten as an initial value problem:

where the linear operator $\mathcal{A}:D(\mathcal{A})\longrightarrow \mathcal{H}$ is defined by

The state space of $\Phi$ is the Hilbert space

equipped with the inner product

for all $\Phi = \left(u, \phi, v, \psi \right) ^{T}$ and $\widetilde{ \Phi } = \left(\widetilde{u}, \widetilde{\phi }, \widetilde{v}, \widetilde{\psi } \right) ^{T}$ in $\mathcal{H}$. The domain of $\mathcal{A}$ is

Then, we have the following local existence result.

Theorem 2.3.Assume that (G4) holds. Then for any $\Phi _{0}\in$ $\mathcal{H},$ problem (2.3) has a unique weak solution $\Phi \in C(\mathbb{R}^{+}; \mathcal{H}).$

Proof. First, for all $\Phi \in D\left(\mathcal{A}\right)$, we have

Therefore, $\mathcal{A}$ is a monotone operator.

To show that $\mathcal{A}$ is maximal, we prove that for each

there exists $V = \left(u, \phi, v, \psi \right) ^{T}\in D\left(\mathcal{A} \right)$ such that $\left(I+\mathcal{A}\right) V = G.$ That is,

From $\left(2.6\right) _{1}$ and $\left(2.6\right) _{3},$ we have $\phi =$ $u-g_{1}$ and $\psi = v-g_{3}.$ Thus, $\left(2.6\right) _{2}$ and $\left(2.6\right) _{4}$ give

Now we define, over $\left[H^{1}(\mathbb{R}^{n})\times H^{1}(\mathbb{R}^{n}) \right] ^{2},$ the bilinear form

and over $H^{1}(\mathbb{R}^{n})\times H^{1}(\mathbb{R}^{n}),$ the linear form

It is easy to verify that $B$ is continuous and coercive over $H^{1}(\mathbb{ R}^{n})\times H^{1}(\mathbb{R}^{n})$ and ${L}$ is continuous on $H^{1}(\mathbb{R}^{n})$. Then, Lax-Milgram theorem implies that the equation

has a unique solution $(u, v)\in H^{1}(\mathbb{R}^{n})\times H^{1}(\mathbb{R} ^{n}).$ Hence,

Thus, $V\in \mathcal{H}$.

Using (2.7) for $\widetilde{v} = 0$, we get

The elliptic regularity theory, then, implies that $u\in H^{2}(\mathbb{R} ^{n})$ and, in addition, Green's formula and (2.6)$_{2}$ give

Hence,

Similarly, using (2.7) for $\widetilde{u} = 0$, we get that $v\in H^{2}(\mathbb{R}^{n})$ and

Therefore,

Consequently, $I+\mathcal{A}$ is surjective and then $\mathcal{A}$ is maximal.

Finally, we show that $J:\mathcal{H\rightarrow H}$ is locally Lipchitz. Hence, we estimate

To handle the first term of the right-hand side of (2.10), we use the mean value theorem to get

Therefore, (G4) implies

All terms in (2.12) are estimated in the same manner. As $u, \widetilde{u}, v, \widetilde{v}\in H^{1}(\mathbb{R}^{n}),$ we use Theorem 2.1, Corollary 2.2 and Hölder's inequality, to obtain

Hence

Also, similar estimations yield to

Inserting (2.14), (2.15) in (2.10), we have

Therefore, $J$ is locally Lipchitz. Thanks to the theorems in Komornik ^[28] (See also Pazy ^[29]), the proof is completed.

The energy functional associated to the above system is given by

Consequently, we present the following lemma.

Lemma 2.4.The solution of $\left(1.1\right)$ satisfies,

where $\lambda = \min \left\{ \lambda _{1}, \lambda _{2}\right\}.$

Proof. Multiplying Eq $\left(1.1\right) _{1}$ by $u_{t},$ equation $\left(1.1\right) _{2}$ by $v_{t}$ and integrating over $\mathbb{R}^{n}$ then summing up we get the result.

Finally, we set

for $t_{0} > 0$ and $\beta > 0$ to be chosen later and $L(t)$ is define by

3.   Blow up

In this section we state and prove our main result.

Theorem 3.1. Assume that (G1)–(G3) hold and the initial data

satisfying,

Then the corresponding solution of (1.1) blows up in finite time.

Before the proof of this theorem, we need the following technical lemmas.

Lemma 3.2.Along the solution of (1.1), the functional $L(t)$ defined in (2.20) satisfies,

and

Proof. A direct differentiation of (2.20) yields

and

Lemma 3.3.Along the solution of (1.1), we estimate,

Proof. Multiply Eq (1.1)$_{1}$ by $u$ and Eq (1.1)$_{2}$ by $v$ and integrate by parts over $\mathbb{R}^{n}$ to get

Exploiting (2.17) to substitute for $\int_{\mathbb{R}^{n}}I(u, v)dx$, we have

Lemma 3.4.Along the solution of (1.1), we estimate, for any $\epsilon > 0,$

Proof. By using Young's inequality, we have

and using Cauchy-Schwartz inequality yields

Recalling (3.2), we estimate

Using (2.18) and (3.1), we have

Hence,

Using (2.19) and (3.7), estimation (3.6) becomes

Hence,

Proof of theorem 3.1. By differentiating $F$ in (2.19) twice we get

and

Inserting (3.3) and (3.4) in (3.9), yield

We then define

for $\gamma > 0$ to be chosen properly.

Differentiating $G$ twice we arrive at

where

Using (3.5), the last term of (3.11) takes the form

Recalling (3.10), estimation (3.11) becomes

It is known, from (G2), that $\frac{\rho }{2} > 1.$ Consequently, we choose $\varepsilon$ $<$ $\frac{\rho }{4}$ and$\; \gamma$ so that

Hence, (3.12) becomes

Now, the choice of

will make

Using the fact that

we infer, from (3.13), that

Then for $\beta$ small enough we conclude, from (3.14), that

and

Therefore, $G^{\prime }$ is decreasing. By choosing $t_{0}$ large enough we get

hence $G^{\prime }(0) < 0.$

Finally Taylor expansion of $G$ yields

which shows that $G(t)$ vanishes at a time $t_{m}\leq -\frac{G(0)}{G^{\prime }(0)}.$ Consequently $F(t)$ must become infinite at time $t_{m}.$

4.   Lower bound for the blow-up time

In this section, we estimate a lower bound for the blow-up time through the following theorem.

Theorem 4.1. Assume that (G1)–(G3) hold and the initial data

are with compact support. Assume further $u$ is a solution of (1.1) which blows up at a finite time $T^{\ast }$. Then there exists a positive constant $C$ such that

where $r$ $= \max \left\{ \beta _{i}, \ i = 1, 2, 3, 4\right\} \geq 1$ and

Before we start the proof, we exploit the finite wave speed of propagation to have the following lemmas.

Lemma 4.2. If the initial data $u_{0}, u_{1}are$ with compact support then for any solution $u$ of (1.1), we have

Proof. In Theorem 2.1, if we let $p = 2$ then we have $p^{\ast } = \frac{2n}{n-2}, \quad n\geq 3$ and

Now

where $L > 0$ is such that

and $B(L+t)$ is the ball, with radius $L+t,$ centered at the origin. Using Hölder inequality, we get

or

Hence, the result follows.

Proof of Theorem 4.1. A direct differentiation of $\psi (t)$ yields

Using Young's inequality and (G1), we have

Using (G3), we get

To estimate the last term of (4.1), we have two cases:

Case I: For $\left\vert u\right\vert \leq \left\vert v\right\vert.$

Using Theorem 2.1 and Corollary 2.2, we have

Case II: For $\left\vert u\right\vert \geq \left\vert v\right\vert.$

Similar to case I, we have

Thus, estimation (4.1) becomes

Recalling the fact that $E(t)\leq E(0) < 0$, we deduce, from (2.17), that

Inserting (4.3) in (4.2), gives

Again (2.17) yields

thus, from (4.4), we have

or

If we use the substitution $y = \psi (t)$ and solve (4.5) over $(0, T^{\ast }),$ then we reach

This completes the proof.

Acknowledgments

The authors thank King Fahd University of Petroleum and Minerals (KFUPM) - Interdisciplinary Research Center (IRC) for Construction and Building Materials for their continuous supports. This work is supported by KFUPM under project # SR181024.

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.