# Global Existence and Decay Estimates of Energy of Solutions for a New Class ofp-Laplacian Heat Equations with Logarithmic Nonlinearity

Authors: Salah Mahmoud Boulaaras; Abdelbaki Choucha; Abderrahmane Zara; Mohamed Abdalla; Bahri-Belkacem Cheri
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The present research paper is related to the analytical studies ofp-Laplacian heat equations with respect to logarithmic nonlinearity in the source terms, where by using an efficient technique and according to some sufficient conditions, we get the global existence and decay estimates of solutions.


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## Body

## 1. A Brief History and Contribution

Consider the following nonlinearp-Laplacian problem: equation with logarithmic nonlinearity:
(1)ut−div∇up−2∇u+up−2u=up−2ulnu,x∈Ω,t>0,ux,0=u0x,x∈Ω,ux,t=0,x∈∂Ω,t≥0,where Ω⊂Rn is a bounded domain with smooth boundary ∂Ω, the function u0 is given initial data and exponent p verify
(2)2<p<∞,ifn≤p,2<p<npn−p,ifn>p.In the last few decades, the researchers have shown significant interest in polynomial nonlinear terms in different areas, such as edge detection, viscoelasticity, engineering, electromagnetic, electrochemistry, cosmology, signal processing material science, turbulence, diffusion, physics, and acoustics. Many other problems in applied sciences are also modeled by linear and nonlinear evolutionary partial differential equations [1–13]. Various dynamical systems in physics and engineering are also modeled by using evolutionary differential equations. Many researchers have contributed a lot to provide an outstanding history of the evolutionary differential partial equations related to px-Laplacian such as [13–17].The majority of problems in science are nonlinear, and it is not easy to find its analytical solutions. The physical problems are mostly designed by using higher nonlinear partial differential equations (PDEs). It is found to be very difficult to find the exact or analytical solutions for such problems. However, in the last several centuries, many scientists have made significant progress and adopted different techniques to study the analytical side of the nonlinear PDEs. Through recent years and in the literature on nonlinear PDEs, logarithmic nonlinearity has received much interest from mathematicians and physicists. If we read in recent research, we notice that logarithmic nonlinearity has been entered into nonrelativistic wave equations that describe spinning particles that move in an external electromagnetic field and in the relativistic wave equation for spinless particles (see, for example, [2, 4, 18, 19]). In addition to what we mentioned above, this type of nonlinearity is used in various branches of physics such as optics, nuclear physics, geophysics, and inflationary cosmology (to read about this in detail, see [18–31]). Given all the basic previous meanings in physics, the study of universal solutions of this type of nonlinear logarithms is of great interest on the part of mathematicians.Recently, Wu and Xue in [32] gave the uniformly proof of energy decay of the solution using the multiplier method of the following problem:
(3)utt−div∇up−2∇u−Δut+utq−1ut=up−1u.Moreover, the author in [33] studied the exponential and polynomial decay rate of solutions for seminar problem (3) by applying the inequality of Nakao.On the another handle, for a Laplacian parabolic equation related to the logarithmic in the right-hand side, the authors in [24] gave the analytical side of the following problem:
(4)ut−Δu−Δut=ulnu.Then, in [27], Nhan and Truong studied the global existence, decay together with the blow up the solutions of the following problem:
(5)ut−div∇up−2∇u−Δut=up−2ulnu,where p>2. In addition, in [25], Cao and Liu gave for 1<p<2, the blow up and global boundedness results of problem (5).Most recently, in [14], Piskin et al. studied the p-Laplacian hyperbolic case
(6)utt−div∇up−2∇u+up−2u+ut=up−2ulnu,x∈Ω,t>0.Motivated by the last mentioned papers, especially [14], in this current research, we consider problem (1) with the presence of nonlinear diffusion Δp=div∇up−2∇u, logarithmic nonlinearity up−2ulnu together with a damping term which is an extension of the previous recent analytical study in [14], where the authors considered the hyperbolic case without damping terms. Our goal is to exploit a potential well method for problem (1) in order to obtain global existence and decay estimate of solutions. More precisely, we give the global existence and decay estimates of solutions under some sufficient conditions.


## 2. Preliminaries

In this section, we put the definitions and lemmas that we need in the rest of the paper:(7)up=uLpΩ,u1,s=uW01,pΩ=up+∇up1/p,for 1<p<∞. We denote the positive constants by C and Ci (i=1,2,⋯).We give the function of energy by(8)Et=1p∇upp+1pupp−1p∫Ωlnuupdx+1p2upp.Lemma 1.
Et is a nonincreasing function, fort≥0(9)E′t=−ut2≤0.Proof.
Multiplying equation (1) by ut and using the integration on Ω, we have
(10)−∫Ωdiv∇up−2∇uutdx+∫Ωup−2uutdx+∫Ωututdx=∫Ωup−2ulnuutdx,(11)ddt1p∇upp+1pupp−1p∫Ωlnuupdx+1p2upp=−ut2.
Thus,(12)E′t=−ut2.Lemma 2 (see [5, 14]).
Letu be any function u∈W01,pRn\0. Then, forp>1, μ>0(13)p∫RnuplnuuLpRndx≤μ∫Rn∇updx−nplnpμenLp∫Rnupdx,where
(14)Lp=pnp−1ep−1π−p/2Γn/2+1Γnp−1/p+1p/n.Remark 3.
Letu∈W01,pΩ\0 and by defining ux=0forx∈Rn\Ω, we can write
(15)p∫ΩuplnuuLpΩdx≤μ∫Ω∇updx−nplnpμenLp∫Ωupdx.Lemma 4 (see [27]).
Letϑ>0. Therefore, we can easy give the following result:
(16)logs≤Csϑ,∀s∈1,∞, such as C=e−1/ϑ.Remark 5.
According to Lemma4, we have
(17)splogs≤Csp+ϑ,s∈1,∞.Lemma 6 (see [34]).
(i)
For all functionu∈W01,pΩ, we have(18)uq≤Bq,p∇up,for every q∈1,∞ if n≤p, and 1≤q≤np/n−p if n>p. We choose constant Bq,p related only on Ω,p and q. Denote Bp,p by Bp.(ii)
For everyu∈W01,pΩ,p≥1 with r≥1, we get(19)uq≤C∇upαur1−α,where C>0,
(20)α=1r−1q1n−1p+1r−1,and we have the following:
(i)
Forp≥n=1,r≤q≤∞(ii)
Forn>1 and p<n,q∈r,np/n−p if r≤np/n−p and q∈r,np/n−p if r≤np/n−p(iii)
Forp=n>1,r≤q<∞(iv)
Forp>n>1,r≤q≤∞


## 3. Result of the Global Existence

We give in this section the proof of the global existence for (1). First, putting the following functionals:
(21)Ju=1p∇upp+p+1p2upp−1p∫Ωlnuupdx,(22)Iu=∇upp+upp−∫Ωlnuupdx.Hence, (21) and (22) give
(23)Ju=1pIu+1p2upp,and we have
(24)Eu=Ju.As in [35], the potential depth of the well is given as
(25)0<d=infusupλ≥0Jλu:u∈W01,pΩ,upp≠0,(26)0<d=infu∈ℵJt.Hence, two sets can be assigned, the first stableW and the second V unstable by
(27)W=u∈W01,pΩ:Ju<d,Iu>0∪0,V=u∈W01,pΩ:Ju<d,Iu<0.Lemma 7.
Letu be all function u∈W01,pΩ\0,upp≠0 and let λ=Jλu. Hence, we have
(i)
limλ→∞gλ=−∞,limλ→0+gλ=0(ii)
Iλu=λg′λ>0,0≤λ<λ∗,=0,λ=λ∗,<0,λ<λ∗<∞,
where(28)λ∗=exp∇upp+upp−∫Ωlnuupdxupp.Proof.
(i)
Fromgλ which we get(29)gλ=Jλu=1pλ∇upp+p+1p2λupp−1p∫Ωlnλuλupdx=λpp∇upp+λppp+1p−lnλupp−λpp∫Ωlnuupdx.
According toupp≠0, we find limλ→∞gλ=−∞, and limλ→0gλ=0.(ii)
From the derivative ofgλ, we get(30)g′λ=ddλJλu=λp−1∇upp+1−lnλupp−∫Ωlnuupdx.
There exists a uniqueλ∗ verify d/dλJλuλ=λ∗, by taking
(31)λ∗=exp∇upp+upp−∫Ωlnuupdxupp.
Of course, we note that the recent property is the result of the following:(32)λdJλudλ=λg′λ=Iλu.
Thus, we have the desired results such that(33)Iλu=λg′λ>0,0≤λ<λ∗,=0,λ=λ∗,<0,λ<λ∗<∞.Lemma 8.
For everyu∈W01,pΩ\0 and l=en+p2/p2p2/nLpn/p2, we get
(i)
If0<up<l, then Iu>0(ii)
IfIu<0, then up>l(iii)
IfIu=0, then up≥lProof.
According to inequality of logarithmic Sobolev, it can be found(34)Iu=∇upp+upp−∫Ωlnuup+lnupupdx≥∇upp+1−lnupupp−μp∫Ω∇updx−np2lnpμenLp∫Ωupdx≥1−μp∇upp+1−lnup+np2lnpμenLpupp.
Selectingμ=p in (34) gives
(35)Iu≥1−lnup+np2lnp2enLpupp.
Thus, we have(i)
If0<up<l, then Iu>0 using the last inequality(ii)
Suppose thatIu<0. This is due to (35), and it(36)up≥en+p2/p2p2nLpn/p2=l(iii)
Similar to the proof of (ii), we prove (iii)
As for functionalJ, it represents the Nehari manifold
(37)ℵ=u∈W01,pΩ\0:Iu=0.
Using Lemma7 in order to prove that ℵ is an unempty set, consider that if u∈ℵ, we obtain
(38)Ju=1p2upp.
We use (23). Further, it proves that J is coercive with respect to ℵ. In addition, if we give Ω1 and Ω2 such that
(39)Ω1=x∈Ω:ux<1,Ω2=x∈Ω:ux≥1.
From Remark5, we can get that
(40)∫Ωuplnudx≤∫Ω1uplnudx+∫Ω2uplnudx≤C∫Ω2up+ζdx≤Cup+ζp+ζ,where ζ>0. Under Lemma 6, we get
(41)∫Ωlnuupdx≤Cup+ζp+ζ≤C∇upαp+ζup1−αp+ζ,where
(42)α=1p−1p+ζ1n−1p+1p−1=nζpp+ζ.
Choosingζ<p2/n, we obtain
(43)αp+ζ<p.
By using Young’s inequality together with (41), we get
(44)∫Ωuplnudx≤ε∇upp+Cεuppβ,where ε>0 and β=1−αp+ζ/p−αp+ζ>1. As u∈ℵ, by (22) and (44), we get
(45)upp+∇upp=∫Ωuplnudx,upp+∇upp≤ε∇upp+Cεuppβ,∇upp≤ε∇upp+Cεuppβ,1−ε∇upp≤Cεuppβ.
Selectε<1. Then, combining (38) and (44), we find
(46)Jt=1p2upp≥Cε∇upp1/β.
Hence, the coercivity ofJ on ℵ.Lemma 9.
(i)
The depth of the potential well is given by(47)d=infu∈ℵJu=infsupλ≥0Jλu:u∈W01,pΩ/0,upp≠0(ii)
d admis a positive lower bound, given by(48)d≥1p2en+p2/pp2nLpn/p=lpp2=K,where Lp is given as in Lemma 2(iii)
There exists a positive functionu∈ℵ, verify Ju=dProof.
(i)
According to Lemma7, it implies that for every u∈W01,pΩ\0, there exists a λ∗, verify Iλ∗u=0, that is λ∗u∈ℵ. Using (47) gives(49)Jλ∗u≥infu∈ℵJu=d.
From Lemma7, the maximizer of Jλu is exact λ∗, such that
(50)supλ≥0Jλu=Jλ∗u=1pIλ∗u+1p2λ∗upp=1p2λ∗upp.
By the combination of (50) and (49), we find
(51)infu∈W01,pΩ/0supλ≥0Jλu=infu∈W01,pΩ/0Jλ∗u≥d.
So that, asu∈W01,pΩ\0, we have d≠0. And if u∈ℵ by (30), we obtain that λ∗ is the only critical point in 0,∞ of the mapping gλ. Therefore,
(52)supλ>0Jλu=Ju,for any u∈ℵ. Then,
(53)infu∈W01,pΩ/0supλ>0Jλu≤infu∈ℵsupλ>0Jλu=infu∈ℵJu=d.
By (51) and (53), (i) is obtained.
(ii)
From Lemma7, ∀u∈W01,pΩ\0, we get Iλ∗t=0. Lemma 8 gives(54)λ∗up≥en+p2/p2p2nLpn/p2=l.
By using (50) and (54), we get
(55)supλ>0Jλu≥lpp2=K.
According to (i), we find thatd≥K.(iii)
Consider the minimize sequenceukk∞⊂u∈ℵ for J, verify(56)limk→∞Juk=d.
Hence, we haveukk∞⊂u∈ℵ is also a minimizing sequence for J due to uk⊂u∈ℵ and Juk=Juk. For this, we can suppose that uk>0 a.e. Ω for any k∈u∈ℵ.
From it, we note thatJ is coercive on u∈ℵ; in other words, ukk∞ is bounded in W01,pΩ. Since W01,pΩ°LpΩ is compact embedding, ∃u is a function and a subsequence of ukk∞, still given by ukk∞, such that
(57)uk→uweaklyinW01,pΩ,uk→ustronglyinLpΩ,ukx→uxa.e.inΩ.
Hence,u≥0 on Ω and
(58)Jt=1p∇upp+p+1p2upp−1p∫Ωlnuupdx≤limk→∞infp+1p2upp+1p∇upp−1p∫Ωlnuupdx=limk→∞infJuk=d.
We apply Lebesgue dominated convergence theorem and weak lower semicontinuity.
Asuk∈u∈ℵ, we have uk∈W01,pΩ\0 and Iuk which implies
(59)ukp≥en+p2/p2p2nLpn/p2=l.
According to Lemma8, we have up≠0 converge strongly in LpΩ; that is to say, that u∈W01,pΩ\0. Moreover, using weak lower continuity, we find
(60)Iu=upp+∇upp−∫Ωlnuupdx≤limk→∞infupp+∇upp−∫Ωlnuupdx=limk→∞infIuk=0.
As a final stage of proof (iii), we prove thatIu=0. If this is false, we get Iu<0; hence, by Lemma 7, ∃λ∗<1 which verifying Iλ∗u=0. Further, we find
(61)0<d≤Jλ∗u=1p2λ∗upp≤λ∗pp2limk→∞infukpp=λ∗plimk→∞infJuk=λ∗pd<d.
And it produces a stark contrast. Meaning that the proof of Lemma9 has ended.Definition 10.
We say that functionut represents a weak solution to problem (1) on Ω×0,T, if
(62)u∈C0,T;W01,pΩ∩C10,T;L2Ω,ut∈L∞0,T;L2Ωsatisfies
(63)∫Ω∇up−2∇u∇wdx+∫Ωup−2uwxdx+∫Ωutwxdx=k∫Ωlnux,tup−2x,twxdx,∀w∈H01Ω,ux,0=u0x.Lemma 11.
Letu0∈W01,pΩ\0 and l=en+p2/p2p2/nLpn/p2. Suppose that 0<E0<lp/p2<d.(i)
Ifu0∈W, then u∈W for 0≤t≤T(ii)
Ifu0∈V, then u∈V for 0≤t≤T,such that T is the maximum time of existence of ut.Proof.
(i)
We putT is the maximum time of existence of solution u. From (24) combined with (47), we find(64)Ju≤Ju0<d,∀t∈0,T.
Then, we haveut∈W for every t∈0,T. If it is false, hence ∃t0∈0,T verify ut0∈∂W, we get either Iu0=0 and Δu0≠0 or (b) Ju0=d.
According to (64), (b) is impossible, that is, Iu0=0 and Δu0≠0. But it is ∃Ju0≥d if 0<d=infu∈ℵJu. From this, we have a stark contrast, ut∈W is obtained for ∀t∈0,T.(ii)
In the same way, we prove case (ii)Theorem 12.
Consideru0x∈W01,pΩ\0. If Iu0>0 and E0<d or u0pp=0. Therefore, problem (1) admits a weak global solution ut∈L∞0,∞;W01,pΩ\0,utt∈L∞0,∞;L2Ω.Proof.
Consider the orthogonal basiswjj=1∞ of the “separable” space W01,pΩ which is orthonormal in L2Ω. Let the following subspace Vm on the finite dimensional
(65)Vm=spanw1,w2,⋯,wm,where the projections of the initial data be defined by
(66)u0mx=∑j=1majwjx→u0inH02Ω,for all j=1,2,⋯,m.
Now, we can see the approximated solutions of (1) as in the following form
(67)umx,t=∑j=1mhjmtwjx,of the approximate problem in Vm(68)∫Ω∇ump−2∇um∇wdx+∫Ωump−2umwxdx+∫Ωutmwxdx=k∫Ωump−2x,tlnumx,twxdx,w∈Vm,um0=u0m=∑j=1mu0,wjwj.
It produces an ordinary differential equation system (ODE) made up of unknown functionshjmt. Starting from the standard theory of existence, there are functions
(69)hj:0,tm→R,j=1,2,⋯,m,which verify (68) in a maximal interval 0,tm,0<tm≤T. Next, we prove that tm=T and that the local solution is uniformly bounded independent of m and t. For this purpose, let us replace w by utm in (68) and integrate by parts, we get
(70)ddtEmt=−utm2≤0,such as
(71)Emt=1p∇umpp+p+1p2umpp−1p∫Ωumplnumdx.
Integrating (70) from 0 to t, and using (24), we obtain
(72)Jum+∫0tusm2ds=Em0.
According to (68), with m→∞, we find Em0→E0. We select m large enough; we find
(73)Jum+∫0tusm2ds<d.
Hence, by (23), we have
(74)Ju=1pIu+1p2upp.
Byu0∈W,(75)Jum0=E0;we select m large enough and 0≤t<∞; we find um0∈W. By (24) and Lemma 11, by picking m large enough and 0≤t<∞, we get mt∈W. Further, according to (24) and (21), we obtain
(76)1p∇umpp+p+1p2umpp−1p∫Ωumplnumdx+∫0tusm2ds<d,where 0≤t<∞. By choosing m large enough and 0≤t<∞ (76), we get
(77)∇umpp<pd,umpp<p2p+1d,∫0tusm2ds<d.
According to Remark5, we find
(78)∫Ωumplnumdx≤∫Ω1umplnumdx+∫Ω2umplnumdx≤C∫Ω2ump+ζdx≤Cump+ζp+ζ,where ζ is pick satisfying p+ζ<np/n−p as p<n and ζ>0 as p≥n and Ω1=x∈Ω:umx<1 and Ω2=x∈Ω:umx≥1.
Applying the embedding theorem, Lemma6 and Young’s inequality, gives from (78):
(79)∫Ωlnumumpdx≤Cump+ζp+ζ≤C∇umupαp+ζump1−αp+ζ≤ε∇umupp+Cεumpp1−αpp+ζ/pp−αp+ζ≤Cε∇umupp.
Therefore, we choose0<ζ for p>αp+ζ, where ε∈0,1 with
(80)α=1p−1p+ζ1n−1p+1p−1,1−αpp+ζp−αp+ζ>1.
Using (79) and (76), for 0≤t<∞, we find
(81)∫Ωlnumumpdx<Cεpd.
Hence, we get(82)umisuniformlyboundedinL∞0,∞;W01,pΩ,utmisuniformlyboundedinL∞0,∞;L2Ω.
Using the integration on (68), we get for 0≤t<∞(83)∫Ωumwsdx=∫Ωu0wsdx+∫0t∫Ωlnumkump−1wsdxds−∫0t∫Ω∇ump−2s∇ums∇wsdxds−∫0t∫Ωump−2sumswsdx.
Further, after passing through the limit in (ref 4030), we arrive at the weak solutionlefturight to the problem (ref 300). According to the initial data in (ref 300), we conclude that ux,0=u0 in W01,p.


## 4. Decay of Solution

In this section, by using the Lyapunov functional, we show the decay of solution to (1).First, we define the Lyapunov functional by(84)Lt=Et+ε2∫Ωu2dx,where ε>0. We will prove the equivalence between Lt and Et.Lemma 13.
Forε>0 small enough, we have
(85)β1Lt≤Et≤β2Lt,where β1,β2>0.
We findL~E by choosing ε small enough.Theorem 14.
Letu0∈V. Assume further 0<E0<p+1/p2μlp<d, where
(86)l=en+p2/p2p2nLpn/p2,μn−p2/npe1−pn+p2pnLp1−p/p<μ<pβ−p+βC∗β−p;hence, ∃c1,c2>0 satisfies
(87)0<Et≤c1e−c2t,t≥0.Proof.
A differentiation ofLt and equation (1) gives
(88)L′t=E′t+ε∫Ωuutdx=−ut2−ε∇upp+upp+ε∫Ωlnuupdx.
Adding and subtractingεβEt into (88) (β>0), we obtain
(89)L′t=−ut2+εβ−pp∇upp+εβ−ppupp+ε1−βp∫Ωlnuupdx+1p2εβupp−εβEt≤−ut2+εβ−pp1+C∗βpβ−p∇upp+εβ−ppupp+ε1−βp∫Ωlnuupdx.
Using the inequality of logarithmic Sobolev together withupp≤C∗∇upp(C∗>0) gives
(90)L′t≤−ut2+εβ−pp1+C∗βpβ−p∇upp+εβ−ppupp+ε1−βp∫Ωlnuupdx−εβEt≤−εβEt−ut2+εβ−pp1+C∗βpβ−p−μp∇upp−εβ−pplnup−np2lnpμenLp+1upp.
Noting that0<β<p and using (21) and Theorem 12, we find
(91)lnup≤lnp2p+1Ju≤lnp2p+1Et≤lnp2p+1E0≤lnμlp=lnμen+p2/ppμnLpn/p.
Byμ satisfying
(92)μn−p2/npe1−pn+p2pnLp1−p/p<μ<pβ−p+βC∗β−p,we guarantee
(93)1+C∗βpβ−p−μp>0,(94)lnup−np2lnpμenLp+1>0;then, we obtain
(95)L′t≤−εβEt−ut2.
Hence, inequality (95) becomes
(96)L′t≤−εβEt.
According to (85), we get
(97)L′t≤−εββ2Lt.
Settingc2=εββ2>0 and integrating (97) yield
(98)Lt≤c1e−c2t.
Finally, by (85), we obtain (87). This is the end of the proof.


## 5. Conclusion

As mentioned earlier in the introduction, the majority of problems in science are nonlinear and their analytical solutions are not easy to find, and most physical problems mostly use higher nonlinear partial differential equations (PDEs). It has been found to be extremely difficult to find accurate or analytical solutions to such problems. However, in the past several centuries, many scientists have made great progress and adopted various techniques to study the analytical side of these famous problems, and nonlinear logarithmic has also received much attention from physicists and mathematicians. Log nonlinearity was introduced into the relativistic wave equation describing spinning particles moving in an external electromagnetic field and in the relativistic wave equation (see, for example, [1–3, 6, 14, 18, 19, 29, 36, 37]); in this contribution, under some sufficient initial and boundary conditions, we have studied the analytical side of p-Laplacian heat equations with respect to logarithmic nonlinearity in the right-hand side, where the global existence and decay estimates of weak solutions are proved. In the next work, we extend our recent work to the coupled system for this important problem. Also, some numerical examples will be given in order to ensure the theory study by using some famous algorithms which are presented in [38, 39].



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