Generalized $(\alpha_s, \xi, \hbar, \tau)$-Geraghty contractive mappings and common fixed point results in partial $b$-metric spaces

1.   Introduction

The biharmonic equation, besides providing a benchmark problem for various analytical and numerical methods, arises in many practical applications. For example, the bending behavior of a thin elastic rectangular plate, as might be encountered in ship design and manufacture, or the equilibrium of an elastic rectangle, can be formulated in terms of the two-dimensional biharmonic equation, e.g., Timoshenko & Woinowsky-Krieger ^[1]. Also, Stokes flow of a viscous fluid in a rectangular cavity under the influence of the motion of the walls, can be described in terms of the solution of this equation, e.g., Pan and Acrivos (1967), Shankar ^[2], Srinivasan ^[3], Meleshko ^[4] or Shankar and Deshpande ^[5]. A more recent application of the biharmonic equation has been in the area of geometric and functional design, where it has been used as a mapping to produce efficient mathematical descriptions of surfaces in physical space, e.g., Sevant et al. ^[6] and Bloor and Wilson ^[7]. Interest in solutions of the biharmonic equation and their mathematical properties go back over 130 years, and comprehensive reviews of this work have been given by Meleshko ^[8,9]. In his review article, he concentrates upon the method of superposition in which the solution is described in terms of a sum of separable solutions of the biharmonic equation. In another work, Meleshko ^[4] obtained some results for Stokes flow in a rectangular cavity in which the solution is based upon the sum of terms consisting of the product of exponential and sinusoidal functions, where the coefficients in the series are determined from the requirement that the prescribed boundary conditions are satisfied, and Meleshko ^[10] described the work which has been done in trying to solve this problem, e.g., Meleshko and Gomilko ^[11]. Other physical phenomena like flows of electro-rheological fluids, fluids with temperature dependent viscocity, filtration processes through a porous media, image processing and thermorheological fluids give rise to mathematical models of hyperbolic, parabolic and biharmonic equations with variable exponents of nonlinearity. More details can also be found in references ^[12,13]. Recently, the hyperbolic equations with nonlinearities of variable exponents type had received a considerable amount of attention. We refer the reader to ^{[14,15,16,17]} and the references therein. Only few works concerning coupled systems of wave equations in the variable-exponents case have been found in the literature. For examples, Bouhoufani and Hamchi ^[18] obtained the global existence of a weak solution and established decay rates of the solutions, in a bounded domain, of a coupled system of nonlinear hyperbolic equations with variable-exponents. Messaoudi et al. ^[15] studied a system of wave equations with nonstandard nonlinearities and proved a theorem of existence and uniqueness of a weak solution, established a blow-up result for certain solutions with positive-initial energy and gave some numerical applications for their theoretical results. In ^[16], Messaoudi et al. considered the following system

with initial and Dirichlet-boundary conditions (here, $f_{1}$ and $f_{2}$ are the coupling terms introduced in (1.3). The authors proved the existence of global solutions, obtained explicit decay rate estimates under suitable assumptions on the variable exponents $m, r$ and $p$ and presented some numerical tests. In this work, we consider the following initial-boundary-value problem

where $\Omega$ is a smooth and bounded domain of $\mathbb{R}^{n}, \left(n = 1, 2, 3 \right),$ the exponents $m$ and $r$ are continuous functions on $\overline{\Omega}$ satisfying some conditions to be specified later, $\frac{\partial u}{\partial \eta}$ denotes the external normal derivatives of $u$ on the boundary $\partial \Omega$ and the coupling terms $f_{1}$ and $f_{2}$ are given as follows: for all $x\in \overline{\Omega }$ and $\left(u, v\right)\in \mathbb{R}^{2},$

with

where$\ a, b > 0\ $are two positive constants and $p$ is a given continuous function on $\overline{\Omega}$ satisfying the condition $(H.2)$ (below).

2.   Preliminaries

This section presents some material needed to prove the main result. Let $q: \Omega \longrightarrow \left[1, \infty \right)$ be a continuous function. We define the Lebesgue space with a variable exponent by

where

Lemma 2.1. ^[13,19] If $1 < q^{-}\leq q(x) \leq q^{+} < +\infty$ holds then, for any $f \in L^{q(.)}(\Omega),$

where

Lemma 2.2. (Embedding property ^[20]) Let $q: \overline{\Omega} \longrightarrow \left[1, \infty \right)$ be a measurable function and $k \geq 1$ be an integer. Suppose that $r$ is a log-Hölder continuous function on $\Omega,$ such that, for all $x\in \Omega,$ we have

Then, the embedding $W^{k, r(.)}_{0}(\Omega) \hookrightarrow L^{q(.)}(\Omega)$ is continuous and compact.

Throughout this paper, we denote by $\mathcal{V}$ the following space

So, $\mathcal{V}$ is a separable Hilbert space endowed with the inner product and norm, respectively,

where $\left\Vert \Delta w \right\Vert_{k} = \left\Vert \Delta w \right\Vert_{L^{k}(\Omega) }.$

We assume the following hypotheses:

$(H.1)$ The exponents $m$ and $r$ are continuous on $\overline{\Omega }$ such that

and

for all $x\in \overline{\Omega},$ where

$(H.2)$ The variable exponent $p$ is a given continuous function on $\overline{\Omega }$ such that

for all $x\in \overline{\Omega}$.

3.   Existence of weak solution

In this section, we prove the local existence of the solutions of (1.2). For this purpose, we introduce the definition of a weak solution for system (1.2). We multiply the first equation in (1.2) by $\Phi \in C^{\infty}_0(\Omega)$ and the second equation by $\Psi \in C^{\infty}_0(\Omega)$, integrate each result over $\Omega$, use Green's formula and the boundary conditions to obtain the following definition:

Definition 3.1. Let $(u_{0}, v_{0})\in \mathcal{V} \times H^1_0(\Omega), (u_{1}, v_{1})\in L^{2}(\Omega)\times L^{2}(\Omega).$ Any pair of functions $(u, v),$ such that

is called a weak solution of (1.2) on $[0, T),$ if

for a.e. $t \in (0, T)$ and all test functions $\Phi \in \mathcal{V}$ and $\Psi\in H^1_0(\Omega)$. Note that $C^{\infty}_0(\Omega)$ is dense in $\mathcal{V}$ and in $H^1_0(\Omega)$ as well. In addition, the spaces $\mathcal{V}$, $H^1_0(\Omega) \subset L^{m(.)}(\Omega) \cap L^{r(.)}(\Omega)$, under the conditions $(H.1)$ and $(H.2)$.

In order to establish an existence result of a local weak solution for the system (1.2); we, first, consider the following auxiliary problem:

for given $f, g \in L^{2}\left(\Omega \times \left(0, T\right) \right)$ and $T > 0.$

We have the following theorem of existence and uniqueness for Problem $(S).$

Theorem 3.1. Let $n = 1, 2, 3$ and $(u_{0}, v_{0})\in \mathcal{V} \times H^1_0(\Omega), (u_{1}, v_{1})\in H^1_0(\Omega) \times L^{2}(\Omega).$ Assume that assumptions $(H.1)$ and $(H.2)$ hold. Then, the problem $(S)$ admits a unique weak solution on $[0, T).$

Proof. Let $\left\lbrace \omega_{j} \right\rbrace _{j = 1}^{\infty}$ be an orthogonal basis of $\mathcal{V}$ and define, for all $k\geq 1,$ $(u^{k}, v^{k})$ a sequence in $\mathcal{V}_{k} = span \left\lbrace \omega_{1}, \omega_{2}, ..., \omega_{k} \right\rbrace \subset \mathcal{V},$ given by

for all $x\in\Omega$ and $\ t\in (0, T)$ and solves the following approximate problem:

for all $j = 1, 2, ..., k$, with

such that

For any $k\geq 1,$ problem $(S_{k})$ generates a system of $k$ nonlinear ordinary differential equations. The ODE standard existence theory assures the existence of a unique local solution $(u^{k}, v^{k})$ for $(S_{k})$ on $[0, T_{k}),$ with $0 < T_{k} \leq T.$ Next, we have to show that $T_{k} = T, \forall k\geq 1.$ Multiplying $(S_{k})_{1}$ and $(S_{k})_{2}$ by $a'_{j}(t)$ and $b'_{j}(t)$, respectively, and then summing each result over $j = 1, ..., k,$ we obtain, for all $0 < t \leq T_{k},$

and

The addition of (3.4) and (3.5), and then the integration of the result, over $(0, t),$ lead to

Using Young's inequality and the convergence (3.3), then Eq (3.6) becomes, for some $C > 0,$

Using the fact that $f, g \in L^{2}\left(\Omega \times \left(0, T\right) \right)$ and choosing $\varepsilon = \frac{1}{4T},$ we infer

where $C_{T} > 0$ is a constant depending on $T$ only. Consequently, the solution $(u^{k}, v^{k})$ can be extended to $(0, T)$, for any $k\geq 1.$ In addition, we have

Therefore, we can extract two subsequences, denoted by $(u^{l})$ and $(v^{l}),$ respectively, such that, when $l \rightarrow \infty,$ we have

Under the assumptions $(H.1)$ and $(H.2)$ and using similar ideas and arguments as in [^[15], Theorem 3.2, p.6], one can see that

and establish that $(u, v)$ satisfies the two differential equations in $(S),$ on $\Omega \times (0, T).$

To handle the initial conditions, we follow the same procedures as in ^[15], and we easily conclude that $(u, v)$ satisfies the initial conditions. For the uniqueness, Assume that $(S)$ has two weak solutions $(u_{1}, v_{1})$ and $(u_{2}, v_{2}),$ in the sense of Definition 3.1. Let $(\Phi, \Psi) = (u_{1t}-u_{2t}, v_{1t}-v_{2t}),$ then $(u, v) = (u_{1}-u_{2}, v_{1}-v_{2})$ satisfies the following identities, for all $t\in \left(0, T\right),$

and

Integrating (3.8) and (3.9) over $(0, t),$ with $t\leq T,$ we obtain

and

But we have, for all $x\in\Omega, Y, Z \in \mathbb{R}$ and $q(x)\geq 2,$

then, estimates (3.10) and (3.11) yield

Thus, $u_{t}(., t) = v_{t}(., t) = 0$ and $u(., t) = v(., t) = 0, \ \text{for all} \ t \in (0, T).$ Thanks to the boundary conditions, we conclude $u = v = 0$ on $\Omega \times(0, T),$ which proves the uniqueness of the solution. Therefore, $(u, v)$ is the unique local solution of $(S),$ in the sense of Definition 3.1, having the regularity (3.1).

Lemma 3.1. Let $y \in L^{\infty}\left((0, T), \mathcal{V}\right)$ and $z \in L^{\infty}\left((0, T), H^1_0(\Omega)\right)$. Then

Proof. From (1.3) and (1.4), we have, for all $(u, v)\in \mathbb{R}^{2},$

and

Let $y \in L^{\infty}\left((0, T), \mathcal{V}\right)$ and $z \in L^{\infty}\left((0, T), H^1_0(\Omega)\right)$. Applying Young's inequality and the Sobolev embedding, we obtain, for all $t\in(0, T)$ and some $C_{1}, C_{2} > 0,$ the following results:

where $C_{0} = 2 \max \left\lbrace a^{2}, 3b^{2}\right\rbrace > 0.$ By the embeddings, we have for $n = 1, 2,$

●

since $3\leq p^{-} \leq p(x) \leq p^{+} < \infty.$ Therefore, estimate (3.16) leads to

where $C_{1} = C_{0}C_{e}$.

● For $n = 3,$ we use the embedding $H^1_0(\Omega)$ in $L^6(\Omega)$ to obtain (3.17), since $p\equiv 3$ on $\overline{\Omega}.$

So, under the assumption $(H.2),$ we have

and similarly

for all $t\in(0, T).$ Which completes the proof.

Corollary 3.1. There exists a unique $(u, v)$ solution of the problem:

in the sense of Definition 3.1 and having the regularity 3.1.

Proof. A combination of Theorem 3.1 and Lemma 3.1 implies this corollary.

Now, consider the following Banach spaces

equipped with the norm:

and

equipped with the norm:

and define a map $F: A_{T} \times B_{T}: \longrightarrow A_{T} \times B_{T}$ by $F(y, z) = (u, v).$

Lemma 3.2. $F$ maps $D(0, d)$ into itself where

Proof. Let $(y, z)$ be in $D(0, d)$ and $(u, v)$ be the corresponding solution of problem $(R)$ (i.e., $F(y, z) = (u, v))$. Taking $(\Phi, \Psi) = (u_{t}, v_{t})$ in Definition 3.1 and integrating each identity over $(0, t),$ we obtain, for all $t \leq T,$

and

The addition of (3.18) and (3.19) lead to

for all $t\in(0, T).$ Therefore,

where $\gamma = \Vert u_{1} \Vert ^{2}_{2}+ \Vert v_{1} \Vert ^{2}_{2}+\|u_{0} \|_{\mathcal{V}}^{2}+\| \nabla v_{0} \|_{2}^{2}.$ Under the assumption (2.3) and applying Young's inequality and the Sobolev embedding (Lemma 2.2), we obtain for all $t\in (0, T),$

where $\varepsilon, c_{1}, c_{2}$ are positive constants. Likewise, we get

Combining (3.21) and (3.22), yields

By substituting (3.23) into (3.20), we obtain, for some $c_{3} > 0,$

Choosing $\varepsilon$ such that $\varepsilon Tc_{3} = \frac{1}{4}$ and recalling that $\left\Vert (y, z) \right\Vert_{A_{T}\times B_{T}} \leq d,$ for some $d > 1$ (large enough), inequality (3.24) implies

So, if we take $d$ such that $d^{2} > > 4 \gamma_{0}$ and $T \leq T_{0} = \frac{d^{2}-4 \gamma_{0}}{c_{4}d^{3(p^{+}-1)}},$ we find

Therefore,

Thus, $F$ maps $D(0, d)$ to $D(0, d)$.

Lemma 3.3. $F: D(0, d) \longrightarrow D(0, d)$ is a contraction.

Proof. Let $(y_{1}, z_{1})$ and $(y_{2}, z_{2})$ be in $D(0, d)$ and set $(u_{1}, v_{1}) = F(y_{1}, z_{1})$ and $(u_{2}, v_{2}) = F(y_{2}, z_{2}).$ Clearly, $(U, V) = (u_{1}-u_{2}, v_{1}-v_{2})$ is a weak solution of the following system

in the sense of Definition 3.1. So, taking $(\Phi, \Psi) = (U_{t}, V_{t}),$ in this definition, using Green's formula together with the boundary conditions and then, integrating each result over $(0, t),$ we obtain, for a.e. $t \leq T,$

and

Under the condition $(H.2),$ using Hölder's inequality and inequality (3.12), these two estimates give, for $n = 1, 2, 3$,

and

The addition of (3.25) and (3.26) imply

for all $t \in (0, T).$ Now, we estimate the terms:

Using appropriate algebraic inequalities (see ^[21]), we obtain for two constants $C_{1}, C_{2} > 0$ and for all $x\in\Omega$ and $t \in (0, T),$

where

By using Hölder's and Young's inequalities and the Sobolev embedding (Lemma 2.2), we get the following estimate for a typical term in $I_{1}$ and $I_{2},$

since

● $1\leq 3(p^{-}-1) \leq 3(p^{+}-1) < \infty,$ when $n = 1, 2.$

● $1 \leq 3(p^{-}-1) = 3(p^{+}-1) = 6 = \frac{ 2n}{n-2},$ when $n = 3.$

Likewise, we obtain

Since $(y_{1}, z_{1}), (y_{2}, z_{2}) \in D(0, d)$ and $d > 1,$ estimates (3.29) and (3.30) lead to

Hence,

Similarly, a typical term in $I_{3}$ can be handled as follows

since

● $1\leq \frac{3}{2}(p^{-}-1) \leq \frac{3}{2}(p^{+}-1) < \infty,$ when $n = 1, 2.$

● $1 \leq \frac{3}{2}(p^{-}-1) = \frac{3}{2}(p^{+}-1) = 6 = \frac{ 2n}{n-2},$ when $n = 3.$

Therefore,

since $(y_{1}, z_{1}), (y_{2}, z_{2}) \in D(0, d).$ Using the same arguments, a typical term in $I_{4},$ can be estimated as follows:

Case 1: If $n = 1, 2,$ we have $3 \leq p^{-} \leq p^{+} < \infty$. So,

since $(y_{1}, z_{1}), (y_{2}, z_{2})\in D(0, d)$ and $d > 1.$

Case 2: If $n = 3,$ then $p \equiv 3$ on $\overline{\Omega}.$ Hence,

So, for all $t\in (0, T),$ we deduce that

Finally, by substituting (3.31)–(3.33) in (3.28), the following can be obtained

for all $t\in(0, T).$ Similarly, we get

Now, we use (3.34) and (3.35) in (3.27) to obtain

Hence, if we take $T$ small enough, we get for, $0 < \gamma < 1,$

Thus,

This proves that $F:D(0, d) \longrightarrow D(0, d)$ is a contraction.

Theorem 3.2. Let $n = 1, 2, 3.$ Under the assumptions $(H.1)$ and $(H.2)$ and for any $(u_{0}, v_{0})\in \mathcal{V} \times H^1_0(\Omega), (u_{1}, v_{1})\in H^1_0(\Omega) \times L^{2}(\Omega)$ the problem (1.2) admits a unique weak solution $(u, v),$ in the sense of Definition 3.1, having the regularity (3.1), for $T$ small enough.

Proof. The above Lemmas and the Banach-fixed-point theorem guarantee the existence of a unique $(u, v) \in D(0, d),$ such that $F(u, v) = (u, v),$ which is a local weak solution of (1.2).

Remark 3.1. From the definitions (1.3) and (1.4), one can easily see that, for all $(u, v)\in \mathbb{R}^{2},$

We, also, have the following results.

Lemma 3.1. ^[22] There exist $C_1, C_2 > 0$ such that, for all $\ x\in \overline{\Omega }$ and $(u, v) \in \mathbb{R}^2$, we have

Corollary 3.2. For all $\ x\in \overline{\Omega }$ and $(u, v) \in \mathbb{R}^2$, we have

where

Now, we introduce the energy functional associated with our problem

for all $t\in [0, T).$ A direct computation implies, for $a.e. \ t \in (0, T),$

4.   Blow-up result

In this section, our goal is to prove that any solution of Problem (1.2) blows-up in some finite time $T^{*}$, if

where

and $c_{*}$ is a positive constant, which comes from the Sobolev embedding.

Remark 4.1. The following well-known inequalities are needed in the proof of the lemmas.

(1) For $\ A, B\geq 0\ {and\ }d \geq 1$, we have

(2) For $z\geq 0, \ 0 < \delta\leq 1\ {and\ } a > 0$, we have

(3) For $X, \ Y\geq 0{, \ }\ \delta > 0\ { and } \ \frac{1}{\lambda }+\frac{1}{\beta } = 1,$ Young's inequality gives

(4) The embedding Lemma 2.2, Hölder's and Young's inequalities and (4.3) imply that

and

Lemma 4.1. For any solution $\left(u, v\right)$ of the system (1.2), with initial energy

and

there exists $\gamma _{2} > \gamma _{1}$ such that

Proof. Let $\gamma = \left(\|\Delta u\|_2^2+\|\nabla v\|_2^2\right) ^{1/2}$, then using (3.39), we have

The use of Lemma 2.1, (4.6) and (4.7) leads to

Combining (4.11) and (4.12), we obtain

For $\gamma \ $in$\ \left[0, \frac{1}{\sqrt{2}c_*} \right]$, one can easily check that

Consequently, we have

Thus, (4.13) reduces to

If we set

then

It is clear that $h$ is strictly increasing on$\ \left[0, \gamma _{1}\right) \ $and strictly decreasing on $\left[\gamma _{1}, +\infty\right)$. Since $E\left(0\right) < E_{1} \ \text{and} \ E_{1} = h\left(\gamma _{1}\right),$ then, we can find $\gamma _{2} > \gamma _{1}\ $such that$\ h\left(\gamma _{2}\right) = E\left(0\right)$. But,

therefore, by (4.14), we get

This implies that$\ \gamma _{0}\geq \gamma _{2}$. Hence, $\gamma _{2}\in \left(\gamma _{1}, \frac{1}{\sqrt{2}c_*}\right].$ To prove (4.10), we assume that there is a $t_0 \in\left[0, T\right)$ such that

Since $\ \left(\|\Delta u\|_2^2+\|\nabla v\|_2^2\right) ^{1/2}$ is continuous and $\gamma_{2} > \gamma _{1}$, $t_0$ can be selected so that

Using (4.14) and the fact that $h\ $ is decreasing on $\left[\gamma _{1}, \frac{1}{\sqrt{2}c_*}\right]$, we obtain

which contradicts the fact that $E\left(t\right) \leq E\left(0\right)$, for all $t\in \left[0.T\right)$. Thus, (4.10) is established.

Lemma 4.2. Let $\mathcal{ H}\left(t\right) = E_{1}-E\left(t\right), \ \mathit{\text{for all}}\ t\in \left[0, \ T\right).$ Then, we have

and

Proof. Using (3.40), (4.8) and (4.11), we have

From the fact that $h(\gamma_1) = \frac{1}{2}\gamma_1 ^{2}-d_*\gamma_1 ^{^{p^{-}+1}} = E_1$, we have

then since $\ \gamma \geq \gamma _{2} > \gamma _{1}$, we obtain

Thus, (4.15) is established. To establish (4.16), we use (4.15) to obtain

which implies,

But $E\left(0\right) = h\left(\gamma _{2}\right) \ $and$\ \gamma\geq \gamma _{2}$, so

Lemma 4.3. There exist $C_{3}, C_{4}, C_{5} > 0$ such that any solution of (1.2) satisfies

and

where $\zeta(u)$ and $\zeta(v)$ are defined in Corollary 3.2.

Proof. We define the following partition of $\Omega$

The properties of $p(.)$ and Hölder's inequality imply that, for some $c_{1} > 0$,

Hence,

Use (4.21) to obtain, for some $c_{2} > 0.$

Recalling (3.38) and (4.15), we deduce that

Therefore,

Similarly, we arrive at

Therefore, (4.18) is established. To establish (4.19), we recall that $p^{-}\geq$ max$\ \left\{ m^{+}, r^{+}\right\}$, to conclude that

Using similar calculations as above, we obtain (4.19) and (4.20).

Lemma 4.4. Let $\mathcal{G}\left(t\right) = \mathcal{H}^{1-\sigma }\left(t\right) +\varepsilon \int_{\Omega }\left(uu_{t}+vv_{t}\right) dx, t > 0,$ where $\varepsilon > 0$ to be fixed later. Then, there exists $\rho > 0$, such that

and hence,

where

Proof. Differentiate $\mathcal{G}$ and use (1.2) to have

By the definition of $\mathcal{H}\ $and $E$, we get

Combining (3.36), (4.25) and (4.26), we obtain

A combination of (4.16) and (4.27) leads to

where $p^{-}-1-2\left(d_*\alpha _{2}^{p^{-}+1}\right) ^{-1}E_{1} > 0$, since$\ \gamma _{2} > \gamma _{1}$.

Now, the last two terms of (4.29) can be estimated by applying (4.5) with $X = \left\vert u\right\vert, \ Y = \left\vert u_{t}\right\vert ^{m\left(x\right) -1}, \ \lambda = m\left(x\right), \ \beta = \frac{m\left(x\right) }{m\left(x\right) -1},$ as follows:

Let $\ \tilde{k}\ $ be a positive constant to be selected later and take $\delta = \left[\tilde{k}\mathcal{H}^{-\sigma }\left(t\right) \right] ^{\frac{1-m\left(x\right) }{m\left(x\right) }}$ to obtain

The properties of $m(x)$ and $\mathcal{H}(t)$ give

where $\tilde{c_{2}} = 1/\left[\mathcal{H}\left(0\right) \right] ^{\sigma \left(m^{+}-1\right)}.$ But $\left[\mathcal{H}\left(0\right) \right] ^{\sigma \left(m\left(x\right) -1\right) }\leq c_{3}, \text{ for all }x\in \Omega,$ where $c_{3} > 0$. So, for some $c_{4} > 0,$ we get

Combining (4.31) and (4.32) to obtain

Applying Similar calculations, we arrive at

Adding (4.33) and (4.34), we have

where $\tilde{\alpha} = \max{\{\frac{m^{+}-1}{m^{-}}\tilde{k}, \frac{r^{+}-1}{r^{-}}\tilde{k}\}}.$ Using (3.43), we have

Hence, (4.35) becomes

Using (3.38) and (4.15), we have

Using the last inequality and (4.19), it can be concluded that

Applying (4.4) with $z = \zeta(u) +\zeta(v), \ a = \mathcal{H}(0), \ \delta = \sigma \left(m^{+}-1\right) +\frac{m^{+}}{p^{-}+1}\ $ and then with$\ \delta = \sigma \left(m^{+}-1\right) +\frac{m^{-}}{p^{-}+1}$, respectively, we get

and

where $\alpha = 1+\frac{1}{ \mathcal{H}(0) }$.

A combination of (4.37)–(4.39) implies that, for some $c_{7} > 0,$

Similar calculations give, for some $c_{8} > 0,$

Using (4.35), (4.40) and (4.41), we obtain, for $c_{9}, c_{10} > 0$,

Inserting (4.42) into (4.29), we have

where $c_{11} > 0$ and $\tilde{R} = \tilde{k}\left(\frac{m^{+}-1}{m^{-}}+\frac{r^{+}-1}{r^{-}} \right)$. Now, we select $\tilde{k}$ large enough so that

where $c_{12} > 0$. Once $\tilde{k}$ is fixed, we select $\varepsilon$ small enough so that

Using the fact that $\mathcal{H}$ is a non-decreasing function, therefore (4.23) is established.

Theorem 4.1. Under the assumptions (4.1) and (4.9), any solution of the system (1.2) blows-up in a finite time.

Proof. Using (4.3) and the definition of $\mathcal{G}$, we have

where $c_{13} = 2^{^{\sigma /\left(1-\sigma \right) }}\max \left\{ 1, \varepsilon ^{1/\left(1-\sigma \right) }\right\}$.

The embedding Lemma 2.2, Lemma 4.2, Hölder's and Young's inequalities give

where $\tau = 2/ \left(p^{-}+1\right)\left(1-2\sigma\right).$

Using (4.15), (3.38) and since $\tau \leq 1$, we get, for some $c_{18} > 0,$

Inserting the last estimate in (4.43), we obtain

Combining (4.23) and (4.45), we deduce that

where $\tilde{c} = \frac{\varepsilon \rho }{c_{16}}$. A simple integration over $(0, t)$ yields

which implies that $\mathcal{G}(t) \longrightarrow +\infty,$ as $t\longrightarrow T^{*},$ where $T^{*} \leq \frac{1-\sigma}{\sigma \tilde{c} \left[\mathcal{G}^{ \frac{\sigma}{ \left(1-\sigma\right) }} \left(0\right) \right] }.$ Consequently, the solution of Problem (1.2) blows-up in a finite time.

5.   Global existence and decay-rate estimates

In this section, we establish the existence of global solutions for initial data in a certain stable set. Then, we show that the decay estimates of the solution energy are exponential or polynomial, depending on the $max\left\lbrace m^{+}, r^{+}\right\rbrace.$

5.1.   Global existence

To state and prove our first result, we introduce the two functionals defined for all $t\in (0, T)$ by

and give the following Lemma.

Lemma 5.1. Under the assumptions $(H.1)$ and $(H.2)$, we suppose that

where

Then,

Proof. From the continuity of $I$ and the fact that $I(0) > 0,$ there exists $t_{k}$ in $\left] 0, T\right)$ such that

We have to show that this inequality is strict.

Recalling (5.1) and (5.2), we have

Combining with (5.4), this gives

From the definition of the energy, we have

for all $t\in \left(0, T\right).$ Consequently,

Thus, the decreasing property of $E$ leads to

On the other hand, from Lemma 2.1 and the Sobolev embedding $H_0^2 (\Omega) \hookrightarrow L^{{p(\cdot) +1}}(\Omega),$ we have

Combining with (5.7), this yields, for all $t\in \left(0, t_{k}\right),$

Therefore,

Similarly, we have

The addition of (5.8) and (5.9) gives

Combining (5.10) with (3.41), we infer that

for all $t\in \left(0, t_{k}\right).$ From the definition of $I,$ this leads to

By repeating the above procedure and using the decreasing property of $E,$ we can extend $t_{k}$ to $T$ and obtain (5.3).

Theorem 5.1. Suppose that all assumptions of Lemma 5.1 are fulfilling. Then, the local solution $\left(u, v\right)$ of the system (1.2) exists globally.

Proof. Substituting (5.5) into (5.6) and thanks to (5.3), it yields

for all $t\in \left(0, T\right).$ Then, we have

for $C_{3} = \max \lbrace 2, \frac{2(p^{+}+1)}{p^{+}-1} \rbrace.$ This means that the norm in (5.12) is bounded independently of $t$. Therefore, the solution $\left(u, v \right)$ exists globally.

5.2.   Decay-rate estimates

To prove the decay result, we need the following Lemma.

Lemma 5.2. Suppose that the assumptions of Lemma 5.1 hold. Then, there exists a positive constant $C_{4}$, such that the global solution $\left(u, v\right)$ satisfies

Proof. The result is immediate by replacing $p$ with $m$ and $r$ in (5.8) and (5.9), respectively, and by recalling (5.12).

Theorem 5.2. Under the assumptions of Lemma 5.1, the solution of the system (1.2) satisfies the following decay estimates, for all $t\geq 0,$

where $\alpha = max \ \lbrace m^{+}, r^{+}\rbrace$ and $k, w > 0$ are two positive constants.

Proof. Multiplying (1.2)$_1$ by $u\left(t\right) E^{\eta}\left(t\right)$ and (1.2)$_2$ by $v\left(t\right) E^{\eta}\left(t\right)$ and then, integrating each result over $\Omega\times\left(s, T\right)$, for $s \in (0, T)$ and $\eta \geq 0$ to be specified later, we arrive at

and

Green's formula and the boundary conditions lead to

and

Adding and subtracting the following two terms

to (5.15) and (5.16), respectively, and recalling (5.11), we arrive at

Now, by exploiting the formula:

estimate (5.17) yields

Next, we handle the terms $I_{i}, i = \overline{1, 4}$ and denote by $C$ a positive generic constant.

● First, applying Young's and Poincaré's inequalities, we obtain

By (5.12), this gives

● Concerning the second term, we have

Again, by (5.12) and the inequalities of Young and Poincaré, we get

and likewise

Therefore,

● For the third term, we apply Young's inequality (as in (4.30)) to obtain, for some $\varepsilon > 0,$

Invoking Lemma 5.2 and recalling (3.40), yields

● Now, we handle $I_{4},$ as follows:

We claim that

Since $2 \leq \tilde{\alpha} \leq m(.) \leq \alpha$ on $\Omega,$ we obtain

where

Therefore,

Three cases are possible:

$(1)$ if $\alpha = \tilde{\alpha} = 2 \ (m(x) = r(x) = 2,$ on $\Omega$), then

$(2)$ if $\alpha > 2$ and $\tilde{\alpha} = 2$, we exploit Young's inequality with

to find

So, for $\eta = \frac{\alpha}{2}-1$, we get

Also, in this case, we have

By inserting (5.24) and (5.25) into (5.23), we infer that $J_{1}$ (and similarly $J_{ 2}$) satisfies (5.22).

$(3)$ if $\alpha > \tilde{\alpha} > 2$, we apply Young's inequality with

to obtain

But $\eta \tilde{\alpha}/ \left(\tilde{\alpha}-2\right) = \eta+1+\left(\alpha-\tilde{\alpha}\right) /\left(\tilde{\alpha}-2\right),$ then

The addition of (5.24) and (5.26) leads to (5.22).

We conclude that the claim is true for any $\alpha \geq \tilde{\alpha} \geq 2.$ Therefore,

Now, substituting (4.22)–(5.21) and (5.27) into (5.18), we get

with $\eta = \frac{\alpha}{2}-1.$ So,

Choosing $\varepsilon$ small enough, we obtain

Letting $T\longrightarrow \infty,$ it yields

Applying Komornik's lemma ^[23], we get the desired decay estimates.

6.   Conclusions

We considered a coupled system of Laplacian and bi-Laplacian equations with nonlinear damping and source terms of variable-exponents nonlinearities. We gave a detailed proof of the local existence using Faedo-Galerkin method and Banach-fixed-point theorem. We also showed that the solutions with positive-initial energy blow-up in a finite time. Furthermore, we proved a global existence theorem, using the Stable-set method and established a decay estimate of the solution energy, by Komornik's integral approach.

Acknowledgment

The authors would like to acknowledge the support provided by King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia. The support provided by the Interdisciplinary Research Center for Construction & Building Materials (IRC-CBM) at King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia, for funding this work through Project No. INCB2205, is also greatly acknowledged.

Conflict of interest

The authors declare that there is no conflict of interest.