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Research article

Fractional transportation problem under interval-valued Fermatean fuzzy sets

  • Received: 15 June 2022 Revised: 09 July 2022 Accepted: 13 July 2022 Published: 25 July 2022
  • MSC : 90C32, 90C70

  • The concept of an interval-valued Fermatean fuzzy set (IVFFS), an extension of Fermatean fuzzy sets, is a more resilient and reliable tool for dealing with uncertain and incomplete data in practical applications. The purpose of this paper is to define a triangular interval-valued Fermatean fuzzy number (TIVFFN) and its arithmetic operations. Fractional transportation problems (FTPs) have important implications for cost reduction and service improvement in logistics and supply management. However, in practical problems, the parameters in the model are not precise due to some unpredictable factors, including diesel prices, road conditions, weather conditions and traffic conditions. Therefore, decision makers encounter uncertainty when estimating transportation costs and profits. To address these challenges, we consider a FTP with TIVFFN as its parameter and call it an interval-valued Fermatean fuzzy fractional transportation problem (IVFFFTP). A new method for solving this IVFFFTP is proposed without re-transforming the original problem into an equivalent crisp problem. Illustrative examples are discussed to evaluate the precision and accuracy of the proposed method. Finally, the results of the proposed method are compared with those of existing methods.

    Citation: Muhammad Akram, Syed Muhammad Umer Shah, Mohammed M. Ali Al-Shamiri, S. A. Edalatpanah. Fractional transportation problem under interval-valued Fermatean fuzzy sets[J]. AIMS Mathematics, 2022, 7(9): 17327-17348. doi: 10.3934/math.2022954

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  • The concept of an interval-valued Fermatean fuzzy set (IVFFS), an extension of Fermatean fuzzy sets, is a more resilient and reliable tool for dealing with uncertain and incomplete data in practical applications. The purpose of this paper is to define a triangular interval-valued Fermatean fuzzy number (TIVFFN) and its arithmetic operations. Fractional transportation problems (FTPs) have important implications for cost reduction and service improvement in logistics and supply management. However, in practical problems, the parameters in the model are not precise due to some unpredictable factors, including diesel prices, road conditions, weather conditions and traffic conditions. Therefore, decision makers encounter uncertainty when estimating transportation costs and profits. To address these challenges, we consider a FTP with TIVFFN as its parameter and call it an interval-valued Fermatean fuzzy fractional transportation problem (IVFFFTP). A new method for solving this IVFFFTP is proposed without re-transforming the original problem into an equivalent crisp problem. Illustrative examples are discussed to evaluate the precision and accuracy of the proposed method. Finally, the results of the proposed method are compared with those of existing methods.



    The wave equation with internal and boundary damping, along with a source term, is described by the system:

    {ωttΔω+ψ1(ωt)=F1(ω)onΩ×R+,ω=0onΓ0×R+,ωtη+ψ2(ωt)=F2(ω)onΓ1×R+,ω(x,0)=ω0(x),ωt(x,0)=ω1(x)inΩ. (1.1)

    In this problem, the functions F1 and F2 are nonlinear source terms on the domain ΩRn and the boundary Ω=Γ0Γ1, respectively, where Γ0 and Γ1 are closed and disjoint and meas.(Γ0)>0. The vector η is the unit outer normal to Ω. The functions ω0 and ω1 are given data. The functions ψ1 is a nonlinear damping acting on the domain Ω, while ψ2 is a nonlinear damping acting on the boundary Ω.

    The study of the existence, blow-up, and stability of solutions to wave equations has been extensively explored in previous research. For example, Lasiecka and Tataru [1] studied the following semilinear model of the wave equation with nonlinear boundary conditions and nonlinear boundary velocity feedback:

    {ωtt=Δωχ0(ω),inΩ×R+,ων=˜χ(ωt|Γ1)χ1(ω|Γ1),onΓ1×R+,ω=0,onΓ0×R+,ω(x,0)=ω0(x),ωt(x,0)=ω1(x)inΩ. (1.2)

    Assuming that the velocity boundary feedback is dissipative and the other nonlinear terms are conservative, uniform decay rates for the solutions are derived. Georgiev and Todorova [2] studied system (1.1) with ψ1(ωt)=|ωt|ϑ2ωt, ψ2(ωt)=F2(ω)=0 and F1(ω)=|ω|q2ω, proving global existence for qϑ and a blow-up result when q>ϑ. Levine and Serrin [3] expanded on this by investigating the case of negative energy with ϑ>1. Rivera and Andrade [4] examined a nonlinear wave equation with viscoelastic boundary conditions, showing the existence and uniform decay under certain initial data restrictions. Santos [5] focused on a one-dimensional wave equation with viscoelastic boundary feedback, demonstrating that under specific assumptions on g and g, sufficient dissipation leads to exponential or polynomial decay if the relaxation function follows the same pattern. Vitillaro [6] explored system (1.1) with ψ1(ωt)=F1(ω)=0 and ψ2(ωt)=|ωt|ϑ2ωt and F2(ω)=|ω|q2u, establishing local and global existence under appropriate conditions on the initial data and exponents. Cavalcanti et al. [7] studied the following problem

    {uttΔω+t0g(ts)Δω(s)ds=0,inΩ×R+,ω=0,onΓ1×R+,ωtνt0g(ts)ωtν(s)ds+ψ(ωt)=0,onΓ0×R+, (1.3)

    where ψ:RR is a nondecreasing C1 function such that

    ψ(s)s>0,for alls0

    and there exist Ci>0,i=1,2,3,4, such that

    {C1|s|p|ψ(s)|C2|s|1p,if|s|1,C3|s||ψ(s)|C4|s|,if|s|>1, (1.4)

    where p1. They proved global existence of both strong and weak solutions, along with uniform decay rates, under restrictive conditions on the damping function ψ and the kernel g. After that, Cavalcanti et al. [8] relaxed these conditions on ψ and g, demonstrating uniform stability based on their behavior. Al-Gharabli et al. [9] extended this work by considering a large class of relaxation functions and establishing general and optimal decay results. Messaoudi and Mustafa [10] focused on system (1.3), exploring more general relaxation functions, and achieved a general decay result without assuming growth conditions on ψ, with the results depending on both g and ψ. Cavalcanti and Guesmia [11] analyzed the following hyperbolic problem involving memory terms

    {ωttΔω+F(x,t,ω)=0,inΩ×R+,ω=0,onΓ0,ω+t0g(ts)ωμ(s)ds=0,onΓ1×R+, (1.5)

    showing that under certain conditions, the memory term dissipation is sufficient to ensure system stability. Specifically, they demonstrated that if the relaxation function decays exponentially or polynomially, the solution follows the same decay rate.

    Liu and Yu [12] investigated the following viscoelastic equation with nonlinear boundary damping and source terms

    {ωttΔω+t0g(ts)Δω(s)ds=0,inΩ×R+,ω=0,onΓ1×R+,ωνt0g(ts)ων(s)ds+|ωt|m2=|ω|p2ω,onΓ0×R+, (1.6)

    proving global existence and general decay of energy under suitable assumptions on the relaxation function and the initial data. Al-Mahdi et al. [13] extended this work by considering system (1.1) with modified terms: F1(u)=0, F2(ω)=|ω|q(x)2ω, ψ1(ωt) is replaced by t0g(ts)Δω(s)ds, ψ2(ωt) is replaced by t0g(ts)ωnds+|ωt|ϑ(x)2ωt, proving global existence and establishing general and optimal decay estimates under specific conditions on the relaxation function and variable exponents ϑ(x) and q(x). They also provided numerical tests to validate their theoretical decay results.

    Zhang and Huang [14] studied a nonlinear Kirchhoff equation described by the system:

    {ωttM(||ω||2)Δω+αωt+χ(ω)=0onΩ×R+,ω=0onΓ1×R+,ωη+ψ(ωt)=0onΓ0×R+,ω(x,0)=ω0(x),ωt(x,0)=ω1(x)inΩ, (1.7)

    where Ω is a bounded domain of Rn with a smooth boundary Ω=Γ0Γ1, and α is a positive real constant. The functions M(s),χ(ω),ψ(ωt) are satisfy some conditions, while η represents the unit outward normal vector. Using the Galerkin approximation, Zhang and Huang established the global existence and uniqueness of the solution. They also addressed challenges posed by the nonlinear terms M(s) and ψ(ωt) through a transformation to zero initial data and employed compactness, monotonicity, and perturbed energy method to resolve the problem. Zhang and Ouyang [15] examined a viscoelastic wave equation with a memory term, nonlinear damping, and a source term:

    {|ωt|ρωttΔω+α|ωt|p2ωt+t0g(ts)Δω(s)ds=|ω|q2ωonΩ×R+,ω=0onΓ×R+,ω(x,0)=ω0(x),ωt(x,0)=ω1(x)inΩ, (1.8)

    where Ω is a bounded domain of Rn with a smooth boundary Ω, ρ,α>0, p2, q>2, and g(t) is a positive function that represents the kernel of the memory term. Using the potential well method combined with the Galerkin approximation, they demonstrated the existence of global weak solutions. Additionally, under certain conditions on the damping coefficient and the relaxation function, they established the optimal decay of solutions via the perturbed energy method. They further showed that the solution can blow up for both positive and negative initial energy conditions.

    For further results on wave equations, see the works of Aassila [16], Wang and Chen [17], Zuazua [18], Soufyane et al. [19], Zhang et al. [20].

    There has been increasing interest among researchers in replacing constant exponents with variable exponents, driven by their practical applications [21] and related references. Variable exponents are commonly used in mathematical models and equations, particularly in damping terms, to better represent a system's diverse behaviors or properties. Damping, which helps dissipate energy and regulate a system's response to external forces, can be more accurately modeled using variable exponents. This allows for a more flexible representation of damping effects tailored to the specific characteristics of the system in question.

    Inspired by these studies and the significance of mathematical models involving nonlinear damping and/or source terms with variable exponents, we consider problem (1.1) with ψ1(ωt)=ψ(ωt), F1(ω)=0 and ψ2(ωt)=|ωt|ϑ(x)2ωt, and F2(ω)=|ω|θ(x)2ω.

    More precisely, we consider the following nonlinear wave equation with internal and boundary damping, along with a source term of variable exponent type:

    {ωttΔω+ψ(ωt)=0onΩ×R+,ω=0onΓ0×R+,ωη+|ωt|ϑ(x)2ωt=|ω|θ(x)2ωonΓ1×R+,ω(x,0)=ω0(x),ωt(x,0)=ω1(x)inΩ. (1.9)

    We aim to study the global existence and stability of solutions to problem (1.9). We investigate the interaction between the internal nonlinear frictional damping and the nonlinear boundary damping of variable exponent type. Additionally, we derive general decay rates, including optimal exponential and polynomial decay rates as the special cases.

    This paper is organized into five sections. In Section 2, we introduce the notation and necessary background material. In Section 3, we prove the global existence of the solution to the problem. In Sections 4 and 5, we present technical lemmas and decay results, respectively.

    In this section, we outline some necessary materials for proving our results. Throughout the paper, we denote a generic positive constant by c. We consider the following assumptions:

    (A1) ϑ:Γ1[1,) is a continuous function such that

    1<ϑ1ϑ(x)ϑ2<q:={2(n1)n2,n>2;,n=1,2,

    where

    ϑ1:=essinfxΓ1ϑ(x),ϑ2:=esssupxΓ1ϑ(x).

    (A2) θ:Γ1[1,) is a continuous function such that

    1<θ1θ(x)ϑ2<q:={2(n1)n2, n > 2 ;, n = 1, 2 , 

    where

    θ1:=essinfxΓ1θ(x),θ2:=esssupxΓ1θ(x).

    Moreover, the variable functions ϑ(x) and θ(x) satisfy the log-Hölder continuity condition.

    For more details about the Lebesgue and Sobolev spaces with variable exponents (see [22,23,24]).

    (A3) ψ:RR is a C0 nondecreasing function satisfying, for c1,c2>0,

    s2+ψ2(s)|Ψ1(sψ(s))for all|s|r,c1|s||ψ(s)|c2|s|for all|s|r,

    where Ψ:(0,)(0,) is C1 function which is a linear or strictly increasing and strictly convex C2 function on (0,r] with Ψ(0)=Ψ(0)=0.

    Remark 2.1. Condition (A3) was introduced for the first time in 1993 by Lasiecka and Tataru [1]. Examples of such functions satisfying Condition (A3) are the following:

    (1) If ψ(s)=csq and q1, then Ψ(s)=csq+12 satisfies (A3).

    (2) If ψ(s)=e1s, then (A3) is satisfied for Ψ(s)=s2e2s near zero.

    We define the energy functional E(t) associated to system (1.9) as follows:

    E(t):=12[ωt22+ω22]Γ11θ(x)|ω|θ(x)dx. (2.1)

    Lemma 2.1. The energy functional E(t) satisfies

    ddtE(t)=Γ1|ωt|ϑ(x)dxΩωtψ(ωt)dx0. (2.2)

    Proof. Multiplying (1.9)1 by ωt integrating over the interval Ω, we have

    ΩωωttΩωtΔωdx+Ωωtψ(ωt)dx=0.

    Using integration by parts, we obtain

    Ωωωtt+Ωωt.ωdxΓ1ωtωtηdx+Ωωtψ(ωt)dx=0.

    Now, using (1.9)3, and doing some modifications, we get

    ddt(12Ωω2tdx+12Ω|ω|2dxΓ11θ(x)|ω|θ(x)dx)=Γ1|ωt|ϑ(x)dxΩωtψ(ωt)dx,

    which gives (2.2).

    For completeness, we present the following existence result, which can be established using the Faedo-Galerkin method and the Banach fixed point theorem, similar to the approaches taken in [2,25,26] for analogous problems.

    Theorem 2.1. (Local existence) Given (ω0,ω1)H1Γ0(Ω)×L2(Ω) and assume that (A1)(A3) hold. Then, there exists T>0, such that problem (1.9) has a weak solution

    ωL((0,T),H1Γ0(Ω))Lθ(.)(Γ1×(0,T)),ωtL((0,T),L2(Ω))Lϑ(.)(Γ1×(0,T)).

    In this section, we state and prove a global existence result under smallness conditions on the initial data (ω0,ω1). For this purpose, we define the following functionals:

    J(t)=12ω221θ1Γ1|ωt|θ(x)dx (3.1)

    and

    I(t)=I(ω(t))=ω22Γ1|ωt|θ(x)dx. (3.2)

    Clearly, we have

    E(t)J(t)+12ωt22. (3.3)

    Lemma 3.1. Suppose that (A1)(A3) hold and (ω0,ω1)H1Γ0(Ω)×L2(Ω), such that

    cθ2eEθ222(0)+cθ2eEθ122(0)<1,I(ω0)>0, (3.4)

    then

    I(ω(t))>0,t>0.

    Proof. Since I is continuous and I(ω0)>0, then there exists Tm<T such that

    I(ω(t))0, t[0,Tm];

    which gives

    J(t)=1θ1I(t)+θ122θ1ω22θ122θ1ω22. (3.5)

    Now,

    ω222θ1θ12J(t)2θ1θ12E(t)2θ1θ12E(0). (3.6)

    Using Young's and Poincaré's inequalities and the trace theorem, we get t[0,Tm],

    Γ1|ω|θ(x)dx=Γ+1|ω|θ(x)dx+Γ1|ω|θ(x)dxΓ+1|ω|θ2dx+Γ1|ω|θ1dxΓ1|ω|θ2dx+Γ1|ω|θ1dxcθ2eωθ22+cθ1eωθ12(cθ2eωθ222+cθ1eωθ122)ω22<ω22, (3.7)

    where

    Γ1={xΓ1:|ω(x,t)|<1}andΓ+1={xΓ1:|ω(x,t)|1}.

    Therefore,

    I(t)=ω22Γ1|ω|θ(x)>0.

    Proposition 3.1. Suppose that (A1)(A3) hold. Let (ω0,ω1)H1Γ0(Ω)×L2(Ω) be given, satisfying (3.4). Then, the solution of (1.9) is global and bounded.

    Proof. It suffices to show that ω22+ωt22 is bounded independently of t. To achieve this, we use (2.2), (3.2) and (3.5) to get

    E(0)E(t)=J(t)+12ωt22θ122θ1ω22+12ωt22+1θ1I(t)θ122θ1ω22+12ωt22. (3.8)

    Since I(t) is positive, Therefore

    ω22+ωt22CE(0),

    where C is a positive constant, which depends only on θ1 and the proof is completed.

    Remark 3.1. Using (3.6), we have

    ω222θ1θ12E(0). (3.9)

    In this section, we present and prove several essential lemmas for demonstrating the main results.

    Lemma 4.1. The functional defined by

    Δ(t)=Ωωωtdx (4.1)

    satisfies, along the solutions of (1.9),

    Δ(t)12Ω|ω|2dx+Ω|ω|θ(x)dx+cΩω2tdx+cΩψ2(ωt)dx+cΓ1|ωt|ϑ(x)dΓ+cΓ|ωt|2ϑ(x)2dΓ, (4.2)

    where Γ={xΓ1:ϑ(x)<2}.

    Proof.

    Δ(t)=Ωω2tdx+ΩωΔωdxΩωψ(ωt)=Ωω2tdxΩ|ω|2dx+Γ1ωωηdΓΩωψ(ωt)=Ωω2tdxΩ|ω|2dxΓ1ω|ωt|ϑ(x)2ωtdΓ+Γ1ω|ω|θ(x)2ωdΓΩωψ(ωt). (4.3)

    The use of Young's and Poincaré's inequalities and choosing ε1=14cp give

    Ωωψ(ωt)dxε1Ωω2dx+14ε1Ωψ2(ωt)dxcpε1Ω|ω|2dx+14ε1Ωψ2(ωt)dx14Ω|ω|2dx+cpΩψ2(ωt)dx. (4.4)

    Define the following partition of Γ1:

    Γ={xΓ1:ϑ(x)<2},Γ={xΓ1:ϑ(x)2}.

    Now, using Young's and Poincaré's inequalities, we obtain

    Γω|ωt|ϑ(x)2ωtdΓλcp||ω||22+14λΓ|ωt|2ϑ(x)2dΓ, (4.5)

    choosing λ=18cp, then we have

    Γω|ωt|ϑ(x)2ωtdΓ18||ω||22+cΓ|ωt|2ϑ(x)2dΓ. (4.6)

    Using Young's inequality with p(x)=ϑ(x)ϑ(x)1 and p(x)=ϑ(x) so, for all xΩ, we have

    |ωt|ϑ(x)2ωtωε2|ωt|ϑ(x)+Cε2(x)|ωt|ϑ(x),

    where

    Cε2(x)=ε1ϑ(x)2(ϑ(x))ϑ(x)(ϑ(x)1)ϑ(x)1.

    Hence, Young's inequality gives

    Γω|ωt|ϑ(x)2ωtdΓε2Γ|ω|ϑ(x)dΓ+ΓCε2(x)|ωt|ϑ(x)dΓcε2(1+(2θ1θ12E(0))ϑ222)||ω||22+ΓCε2(x)|ωt|ϑ(x)dΓ. (4.7)

    Choosing ε2=18c(1+(2θ1θ12E(0))ϑ222), then Cε2(x) is bounded and noting that ΓΓ1, then we have

    Γω|ωt|ϑ(x)2ωtdΓ18||ω||22+cΓ1|ωt|ϑ(x)dΓ. (4.8)

    By combining the above estimates, the proof is completed.

    Lemma 4.2. Let us introduce perturbed energy functional as follows:

    M(t)=NE(t)+Δ(t)

    satisfies, for all t0 and for a positive constant N,

    M(t)cE(t)cE(t)+cΩ(ω2t+ψ2(ωt))dx+cΓ|ωt|2ϑ(x)2dΓ. (4.9)

    Proof. We establish the proof by means of perturbed energy method. Taking the derivative of M with respect to t, and using the estimates in (4.2), and (2.2), we obtain

    M(t)NΓ1|ωt|ϑ(x)dxNΩωtψ(ωt)dx12Ω|ω|2dx+Ω|ω|θ(x)dx+cΩω2tdx+cΩψ2(ωt)dx+cΓ1|ωt|ϑ(x)dΓ+cΓ|ωt|2ϑ(x)2dΓ. (4.10)

    Choosing N large enough such that ME, and recalling (2.2), therefore the proof of (4.9) is completed.

    Lemma 4.3. If 1<ϑ1<2, then the following estimate holds:

    Γ|ωt|2ϑ(x)2dΓcE(t)cE(t)(E(t))2ϑ12ϑ12cE(t). (4.11)

    Proof. First, we define the following partition:

    Γ1={xΓ:|ωt(t)|1},Γ2={xΓ:|ωt(t)|>1},

    and use the fact that 2ϑ(x)2ϑ(x)2ϑ12ϑ1, and Jensen's inequality to obtain

    Γ|ωt|2ϑ(x)2dΓ=Γ1|ωt|2ϑ(x)2dΓ+Γ2|ωt|2ϑ(x)2dΓ=Γ1[|ωt|ϑ(x)]2ϑ(x)2ϑ(x)dΓ+Γ2|ωt|ϑ(x)+ϑ(x)2dΓcΓ1[|ωt|ϑ(x)]2ϑ12ϑ1dΓ+cΓ2|ωt|ϑ(x)dΓc[E(t)]2ϑ12ϑ1cE(t). (4.12)

    Using Young's inequality, we find that

    [E(t)]2ϑ12ϑ1=(E(t))2ϑ12ϑ12[E(t)]2ϑ12ϑ1(E(t))2ϑ12ϑ12ε(E(t))ϑ12ϑ12CεE(t)(E(t))2ϑ12ϑ12=εE(t)CεE(t)(E(t))2ϑ12ϑ12. (4.13)

    Choosing ε small enough, the proof of (4.11) is completed.

    Remark 4.1. If ϑ12 and since meas(Γ)=0 then

    Γ|ωt|2ϑ(x)2dΓ=0. (4.14)

    Lemma 4.4. Under assumption (A3), the following estimates hold:

    Ωωtψ(ωt)dxcE(t),if   ψ   is linear,  (4.15)
    Ωωtψ(ωt)dxcΨ1(Λ(t))cE(t),if   ψ   is nonlinear,  (4.16)

    where Λ(t) is defined in the proof.

    Proof. Case 1: ψ is linear, then

    cΩ(ω2t+ψ2(ωt))dxcE(t).

    Case 2: ψ is nonlinear, we define the following partition of Ω

    Ω1={xΩ:|ωt|r},Ω2={xΩ:|ωt|r},

    where r is small enough such that

    sψ(s)min{r,ψ(r)},|s|r.

    We also define

    Λ(t)= Ω1ωtψ(ωt)dx.

    Now, using hypothesis (A3) and Jensen's inequality, we get

    Ω1(ω2t+ψ2(ωt))dxΩ1Ψ1(ωtψ(ωt))dxcΨ1(Λ). (4.17)

    In this section, we state and prove the stability result of system (1.9).

    Theorem 5.1. Assume that ϑ12 and ψ is linear. Then

    E(t)κ1eκ2t, (5.1)

    for some positive constants κ1 and κ2.

    Proof. Combining (4.9), (4.15) with (4.14), we obtain,

    M(t)cE(t)cE(t).

    Therefore, M+cEE and a simple integration over (0,t) yields, for some κ1,κ2>0,

    E(t)κ1eκ2t,t0.

    Theorem 5.2. Assume that 1<ϑ1<2 and ψ is linear. Then

    E(t)c(1+t)1α, (5.2)

    where α=2ϑ12ϑ12>0.

    Proof. From (4.9), (4.11) and (4.15), we have

    M1(t)cE(t)cE(t)(E(t))2ϑ12ϑ12, (5.3)

    where M1=M+cEE. Multiply both sides of (5.13) by (E(t))α where α=2ϑ12ϑ12, to obtain

    M2(t)cEα+1(t), (5.4)

    where M2=(E(t))αM1+cEE. Integrating over (0,t) and using the equivalence relation lead to (5.2).

    Theorem 5.3. Assume that ϑ12 and ψ is nonlinear. Then, for some positive constants ϱ1 and ϱ2, we have

    E(t)Ψ11(ϱ1t+ϱ2),t0, (5.5)

    where Ψ1(t)=1t1Ψ2(s)ds and Ψ2(t)=tΨ(ε0t)

    Proof. From (4.9), (4.1) and (4.15), we have

    M(t)cE(t)+cΨ1(Λ(t)). (5.6)

    Now, for ε0<r, using the fact that E0, Ψ>0,Ψ>0 on (0,r], we find that the functional ˜M, by

    ˜M(t):=Ψ(ε0E(t)E(0))M(t)+c0E(t),

    satisfies, for some α1,α1>0,

    α1˜M(t)E(t)α2˜M(t), (5.7)

    and

    ˜M(t)=ε0E(t)E(0)Ψ(ε0E(t)E(0))M(t)+Ψ(ε0E(t)E(0))M(t)+c0E(t)cE(t)Ψ(ε0E(t)E(0))+cΨ(ε0E(t)E(0))Ψ1(Λ(t))+c0E(t). (5.8)

    Let Ψ be the convex conjugate of Ψ in the sense of Young with A=Ψ(ε0E(t)E(0)) and B=Ψ1(Λ(t)), we arrive at

    ˜M(t)cE(t)Ψ(ε0E(t)E(0))+cΨ(Ψ(ε0E(t)E(0)))+cΛ(t)+c0E(t)cE(t)Ψ(ε0E(t)E(0))+cε0E(t)E(0)Ψ(ε0E(t)E(0))cE(t)+c0E(t).

    Consequently, with a suitable choice of ε0 and c0, we obtain, for all t0,

    ˜M(t)cE(t)E(0)Ψ(ε0E(t)E(0))=cΨ2(ε0E(t)E(0)), (5.9)

    where Ψ2(t)=tΨ(ε0t). Since Ψ2(t)=Ψ(ε0t)+ε0tΨ(ε0t), then, using the strict convexity of Ψ on (0,r], we find that Ψ2(t),Ψ2(t)>0 on (0,1]. Thus, with

    Φ(t)=εα1˜M(t)E(0),0<ε<1,

    taking in account (5.7) and (5.9), we have

    Φ(t)E(t), (5.10)

    and then

    Φ(t)cΨ2(Φ(t)),t0.

    Then, a simple integration gives, for some ϱ1,ϱ2>0,

    Φ(t)Ψ11(ϱ1t+ϱ2),t0, (5.11)

    where Ψ1(t)=1t1Ψ2(s)ds. A combination of (5.10) and (5.11) gives (5.5).

    Theorem 5.4. Assume that 1<ϑ1<2 and ψ is nonlinear. Then, for some positive constants ϱ3 and ϱ4, we have

    E(t)χ11(ϱ3t+ϱ4),t0, (5.12)

    where χ1(t)=1t1Ψ2(s)ds, χ2(t)=tχ(ε0t), χ=(G1+Ψ1)1 and G(t)=tϑ12ϑ12.

    Proof. From (4.9) and (4.13), we have

    M(t)cE(t)+(E(t))2ϑ12ϑ1+cΨ1(Λ)(t), (5.13)

    where M=EαM+cEE. Let G(t)=tϑ12ϑ12. Then the last inequality can be written as

    M(t)cE(t)+G1(E(t))+cΨ1(Λ)(t). (5.14)

    Therefore, (5.14) becomes

    M(t)cE(t)+cχ1(ξ(t)), (5.15)

    where χ=(G1+Ψ1)1 and ξ(t)=max{E(t),Λ(t)}. Define the following functional

    K(t):=χ(ε0E(t)E(0))M(t)+c0E(t), (5.16)

    satisfies, for some α2,α3>0,

    α2K(t)E(t)α3K(t). (5.17)

    Combining (5.15) and (5.16), we obtain

    K(t)cE(t)χ(ε0E(t)E(0))+χ(ε0E(t)E(0))χ1(ξ(t))+c0E(t). (5.18)

    Let χ be the convex conjugate of χ in the sense of Young, then

    χ(s)=s(χ)1(s)χ[(χ)1(s)],ifs(0,χ(r)] (5.19)

    and χ satisfies the following generalized Young inequality

    ABχ(A)+χ(B),ifA(0,χ(r)],B(0,r]. (5.20)

    Thus, with A=χ(ε0E(t)E(0)) and B=χ1(ξ(t)), we arrive at

    K(t)cE(t)χ(ε0E(t)E(0))+cε0E(t)E(0)χ(ε0E(t)E(0))cE(t)+c0E(t).

    Choosing c0,ε0 small enough, we get

    K(t)cε0E(t)E(0)χ(ε0E(t)E(0))=cχ2(ε0E(t)E(0)),

    where χ2(t)=tχ(ε0t). Letting

    Y(t)=εα3K(t)E(0),0<ε<1,

    and taking in account (5.7) and (5.9), we have

    Y(t)E(t), (5.21)

    and then

    Y(t)cχ2(Y(t)),t0.

    Then, a simple integration gives, for some ϱ3,ϱ4>0,

    Y(t)χ11(ϱ3t+ϱ4),t0, (5.22)

    where χ1(t)=1t1χ2(s)ds, which finishes the proof.

    Examples 5.1. The following examples illustrate our results:

    (1) If ψ(t)=ct and ϑ(x)=2, then

    E(t)c1ec2t, (5.23)

    which is an exponential decay.

    (2) If ψ(t)=ct and ϑ(x)=234+x, then ϑ1=54 and ϑ2=75, then the energy functional satisfies

    E(t)c(1+t)23. (5.24)

    (3) If ψ(t)=ct2 and ϑ(x)=2+11+x, then ϑ1=52, ϑ2=3 and ψ(t)=ct32. Then,

    Ψ11(t)=(ct+1)2.

    Therefore, we obtain

    E(t)c(1+t)2. (5.25)

    (4) If ψ(t)=ct5 and ϑ(x)=234+x, then ϑ1=54, ϑ2=75 and ψ(t)=ct3. Then,

    χ(s)=(G1+Ψ1)1=(1+1+4s2)3

    and

    χ2(s)=3s1+4s(1+1+4s2)2=3s21+4s+3s21+4s3s23s2+3s22s3s2=cs32.

    Therefore, we obtain

    E(t)c(1+t)13.

    In this work, we consider a nonlinear wave equation with internal and boundary damping and a source term of variable exponent type. We prove the global existence and stability of solutions to this problem problem. We study the interaction between the internal nonlinear frictional damping and the nonlinear boundary damping of variable exponent type. In addition, we establish general decay rates, including optimal exponential and polynomial decay rates as the special cases.

    Adel M. Al-Mahdi: Conceptualization, methodology, formal analysis, writing-original draft; Mohammad M. Al-Gharabli: Formal analysis, validation, writing-reviewing and editing; Mohammad Kafini: Conceptualization, methodology, formal analysis, reviewing. All authors have read and approved the final version of the manuscript for publication.

    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. INCB2402), is also greatly acknowledged.

    This work is funded by KFUPM, Grant No. INCB2402.

    The authors declare no competing interests.



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