# On Local Weak Solutions for Fractional in Time SOBOLEV-Type Inequalities

Authors: Mohaemd Jleli; Bessem Samet
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We consider two fractional in time nonlinear Sobolev-type inequalities involving potential terms, where the fractional derivatives are defined in the sense of Caputo. For both problems, we study the existence and nonexistence of nontrivial local weak solutions. Namely, we show that there exists a critical exponent according to which we have existence or nonexistence.


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## Body

## 1. Introduction and Main Results

We consider the fractional in time Sobolev-type inequalities(1)∂tσ+Δηt,z+Δηt,z1+z2ρ/2ηt,z≤p0,t,z∈0,∞×ℝN,η0,z=η0z,z∈ℝNand
(2)∂tσ+1+Δηt,z+Δηt,z1+z2ρ/2ηt,z≤p0,t,z∈0,∞×ℝN,η0,z=η0z,∂tη0,z=η1z,z∈ℝN,where N≥1, p>1, and ρ>−2. Here, σ∈0,1 and ∂tλ,λ∈σ,σ+1 is the derivative of fractional order λ in the sense of Caputo. Namely, we are concerned with the existence and nonexistence of nontrivial local weak solutions to problems (1) and (2). We shall establish that there exists a critical exponent pc>1 that depends on N and ρ such that if p ∈ (1, pc), then problems (1) and (2) admit no nontrivial local weak solutions (i.e., we have an instantaneous blow-up), while if p∈pc,∞, then the considered problems admit local solutions for some initial values. In the proofs of our results, we use the test function method with some integral estimates. For more details about the test function method and its applications to partial differential equations, we refer to [1–3] and the references therein.The absence of solutions (complete blow-up phenomenon) was observed in [4] for the following elliptic inequality with a singular potential term
(3)−Δη≥η2z2inD′gΩ\0,η≥0a.e.,where Ω is a smooth bounded domain in RN containing 0. In the same reference, an instantaneous blow-up result was obtained for the parabolic analogue of (3), namely
(4)∂tη−Δη≥η2z2inD′0,T×Ω0,η≥0a.e.,Notice that the method in [4] is based on comparison principles. In [5], using the test function method and avoiding the maximum principle, instantaneous blow-up results were obtained for certain classes of elliptic and parabolic inequalities including as special cases (3) and (4). For more results on instantaneous blow-up for nonlinear evolution equations, we refer to [6–9] and the references therein.The investigation of instantaneous blow-up for linear Sobolev-type equations was first considered in [10]. Namely, the following problem was studied
(5)∂tηzz+η=ηzz,η0,z=η0z,ηt,0=ηt,L,L>0.In the limit caseσ=1 and ρ=0, (1) and (2) (with equalities instead of inequalities), reduce, respectively, to
(6)∂t+Δηt,z+Δηt,z+ηt,zp=0,t,z∈0,∞×ℝN,η0,z=η0z,z∈ℝN(7)∂tt+Δηt,z+Δηt,z+ηt,zp=0,t,z∈0,∞×ℝN,η0,z=η0z,∂tη0,z=η1z,z∈ℝN.The nonexistence of local weak solutions to (6) and (7) was considered in [11] by the use of the test function method. When N∈1,2, it was proved that for all p>1, (6) and (7) admit no nontrivial local weak solutions. If N≥3, it was shown that if 1<p≤N/N−2, then (6) and (7) admit no nontrivial local weak solutions, while if P>N/N−2, then local solutions exist. Our aim in this paper is to study the instantaneous blow-up for the fractional in time versions of (6) and (7) with the potential term Vz=1+z2ρ/2.For existence results for stationary problems involving potential terms, see for example [12] and the references therein.Notice that the study of fractional in time Sobolev-type equations was first considered in [13], where the nonexistence of global weak solutions was investigated.Before mentioning our main results, let us define the meaning of solutions to (1) and (2).LetT∈0,∞. Given κ>0 and ω∈C0,T, T>0, we define the fractional integral operators
(8)J0κωt=1Γκ∫0tt−ςκ−1ωςdς,0<t≤Tand
(9)JTκωt=1Γκ∫tTς−tκ−1ωςdς,0<t≤T.Givenωi∈C0,T, i=1,2, one has (see e.g., [14, 15])
(10)∫0TJ0κω1tω2tdt=∫0TJTκω2tω1tdt.Forσ∈0,1, the derivatives of fractional orders σ and σ+1 in the Caputo sense are defined, respectively, by
(11)∂tσωt=J01−σω′t,0<t≤T,ω∈C10,T∂tσ+1ωt=J01−σω′′t,0<t≤T,ω∈C20,T.Using the above notions and property (10), we define local weak solutions to problem (1) as follows.Definition 1.
Letη0∈Lloc1ℝN. We say that η is a local weak solution to (1), if there exists η∈LlocpΛT,N and satisfies
(12)∫ΛT,N1+z2ρ/2ηpξdzdt−∫RNη0zΔJT1−σξ0,zdz≤−∫ΛT,NηΔξdzdt+∫ΛT,NηΔ∂tJT1−σξdzdt,for all ξ∈C∞ΛT,N, ξ≥0, with suppzξ⊂⊂RN. Here, ΛT,N is the product set 0,T×RN. Local weak solutions to problem (2) are defined as follows.Definition 2.
Letη0,η1∈L1locRN. We say that η is a local weak solution to (2), if there exists T∈0,∞ such that η∈LlocpΛT,N and satisfies
(13)∫ΛT,N1+z2ρ/2ηpξdzdt+∫ℝNη0zΔ∂tJT1−σξ0,zdz−∫ℝNη1zΔJT1−σξ0,zdz≤−∫ΛT,NηΔξdzdt−∫ΛT,NηΔ∂ttJT1−σξdzdt,for all ξ∈C∞ΛT,N, ξ≥0, with suppzξ⊂⊂RN and ∂tJT1−σξT,·≡0.Now, we state our results. We first define the (critical) exponent(14)pc=∞ifN∈1,2,N+ρN−2ifN≥3.Theorem 3.
LetN≥1,ρ>−2, and σ∈0,1.
(i)
Ifη0∈L1RNand1<p<pc, then problem (1) admits no nontrivial local weak solution(ii)
IfN≥3andp>pc, then problem (1) admits local solutions for some η0>0Theorem 4.
LetN≥1, ρ>−2, and σ∈0,1.
(i)
Ifη0,η1∈L1ℝNand1<p<pc, then problem (2) admits no nontrivial local weak solution(ii)
IfN≥3andp>pc, then problem (2) admits local solutions for someη0>0andη1≡0The next section contains some preliminary estimates that will be useful in the proofs of our results. Section3 is devoted to the proofs of Theorems 3 and 4.


## 2. Preliminaries

ForS,T∈0,∞, let
(15)αt=1−T−1tm,t∈0,Tand
(16)βz=Fz2S2m,z∈ℝN,where m≫1 (i.e., sufficiently large) is a natural number and F∈C∞R+ satisfies
(17)0≤F≤1,F0,1≡1,F2,∞≡0.Let us introduce the function(18)ξt,z=αtβz,t,z∈ΛT,N.Clearly, one hasξ∈C∞ΛT,N,ξ≥0andsuppzξ⊂⊂ℝN.Lemma 5.
The functionα defined by (15) satisfies the following properties:
(19)JTkαt=m!Γκ+m+1T−mT−tκ+m,t∈0,T,(20)αtJTkαt=m!Γκ+mT−mT−tκ+m−1,t∈0,T,(21)∂ttJTkαt=m!Γκ+m+1T−mT−tκ+m−2,t∈0,T,where κ∈0,1.Proof.
One has(22)JTkαt=1ΓκT−m∫tTς−tκ−1T−ςmdς.Takingτ=T−ς/T−t, the above integral reduces to
(23)JTkαt=1ΓκT−mT−tκ+m∫011−τκ−1τmdτ=1ΓκT−mT−tκ+mΓκΓm+1Γκ+m+1,=m!Γκ+m+1T−mT−tκ+m,which proves (19). Next, (20) and (21) follow by differentiating (19).Lemma 6.
For sufficiently largeS, one has
(24)∫ℝNβz−1/p−11+z2−ρ/2p−1Δβzp/p−1dz≤CSN−2p+p/p−1.Here,C>0 is a constant (independent on S).Proof.
Using(16) and (17), one obtains
(25)∫ℝNβz−1/p−11+z2−ρ/2p−1Δβzp/p−1dz=∫Sl<z<2Slβz−1/p−11+z2−ρ/2p−1Δβzp/p−1dz.On the other hand, an elementary calculation shows that(26)Δβz≤CS−2Fz2S2m−2,S<z<2S.Here and below,C>0 is a constant independent on S, whose value may change from line to line. Hence, one deduces that
(27)∫ℝNβz−1/p−11+z2−ρ/2p−1Δβzp/p−1≤CS−2p/p−1∫S<z<2S1+z2−ρ/2p−1Fz2S2m−2p/p−1dz≤CS−2p/p−1∫S<z<2S1+z2−ρ/2p−1dz≤CS−2p/p−1∫r=S2S1+r2−ρ/2p−1rN−1dr≤CS−2p/p−11+S2−ρ/2p−1SN−1S=CSN−2p+ρ/p−1,which proves the desired result.The following result is obvious.Lemma 7.
ForT>0, one has
(28)∫0Tαtdt=Tm+1.Using (20), one obtains easily the following result.Lemma 8.
ForT>0, one has
(29)∫0Tαt−1/p−1αtJTκαtdt=CT1+κ−1p/p−1,where κ∈0,1.Using (21), the following result follows.Lemma 9.
ForT>0, one has
(30)∫0Tαt−1/p−1∂ttJTκαtdt=CT1+κ−2p/p−1,where κ∈0,1.


## 3. Proofs of the Main Results

Proof of Theorem 10.
(i)
Suppose thatη∈LlocpΛT,N is a nontrivial local wessak solution to (1) for some fixed T∈0,∞. Then, using (12) with ξ is the function defined by (18), one obtains(31)∫ΛT,N1+z2ρ/2ηpξdzdt−∫ℝNη0zΔJT1−σξ0,zdz≤∫ΛT,NηΔξdzdt+∫ΔT,NηΔ∂tJT1−σξdzdt.Next, usingε-Young inequality with 0<ε<1/2, one obtains
(32)∫ΛT,NηΔξdzdt≤ε∫ΛT,N1+z2ρ/2ηpξdzdt+C+∫ΔT,Nξ−1/P−11+z2−ρ/2p−1Δξp/p−1dzdt.Similarly, one has(33)∫ΛT,NηΔ∂tJT1−σξdzdt≤ε∫ΛT,N1+z2ρ/2ηpξdzdt+C∫ΔT,Nξ−1/P−11+z2−ρ/2p−1Δ∂tJT1−σξp/p−1dzdt.It follows from (31), (32), and (33) that
(34)1−2ε∫ΛT,N1+z2ρ/2ηpξdzdt+∫ℝNη0zΔJT1−σξ0,zdz≤C∫ΛT,Nξ−1/p−11+z2−ρ/2p−1Δξp/p−1dzdt+∫ΛT,Nξ−1/p−11+z2−ρ/2p−1Δ∂tJT1−σξp/p−1dzdt.On the other hand, by (18), one has
(35)∫ΛT,Nξ−1/p−11+z2−ρ/2p−1Δξp/p−1dzdt=∫0Tαtdt∫ℝNβz−1/p−11+z2−ρ/2p−1Δβzp/p−1dz.Hence, using Lemmas6 and 7, for sufficiently large S, one obtains
(36)∫ΛT,Nξ−1/p−11+z2−ρ/2p−1Δξp/p−1dzdt≤CTSN−2p+ρ/p−1.Again, by (18), one has
(37)∫ΛT,Nξ−1/p−11+z2−ρ/2p−1Δ∂tJT1−σξp/p−1dzdt=∫0Tαt−1/p−1∂tJT1−σαtdt∫ℝNβz−1/p−11+z2−p/2p−1Δβzp/p−1dz.Therefore, using Lemma6 and Lemma 8 with κ=1−σ, for sufficiently large S, one obtains
(38)∫ΛT,Nξ−1/p−11+z2−ρ/2p−1Δ∂tJT1−σξp/p−1dzdt≤CT1−σp/p−1SN−2p+ρ/p−1.On the other hand,
(39)∫ℝNη0zΔJT1−σξ0,zdz=JT1−σα0∫ℝNη0zΔβzdz.Hence, using (19) (with κ=1−σ) and (26), one deduces that
(40)∫ℝNη0zΔJT1−σξ0,zdz≤CT1−σS−2∫ℝNη0zFz2S2m−2dz.Notice that sinceη0∈L1ℝN, by (17), one deduces from the above estimate that
(41)limS→∞∫ℝNη0zΔJT1−σξ0,zdz=0.Next, it follows from (34), (36), (38), and (40) that
(42)∫0T∫ℝN1+z2ρ/2ηpαtFz2S2mdzdt≤CTSN−2p+ρ/p−1+T1−σp/p−1SN−2−+ρ/p−1+∫ℝNη0zΔJT1−σξ0,zdz.Notice that since1<p<pc, then N−2p+p/p−1<0. Hence, passing to the infimum limit as S⟶∞ in (42), using (17), (41), and Fatou’s lemma, one deduces that
(43)∫0T∫ℝN1+z2ρ/2ηpαtdzdt=0,which contradicts the fact that η is nontrivial.
(ii)
Let(44)ρ+2p−1≤a<N−2(45)ϰ=aN−a−21/p−1.Notice that sincep>pc, the set of values of a satisfying (44) is nonempty. Consider the function
(46)wz=ϰ1+z2−a/2,z∈ℝN.An elementary calculation shows that(47)∆wz=ϰa1+z2−a/2−2N+N−a−2z2.Hence, using (44), (45), and (46), one obtains
(48)−∆wz=1+z2ρ/2wzp=ϰa1+z2−a/2/−2N+N−a−2z2−ϰp1+z2ρ/2−ap/2>ϰaN−a−21+z2−a/2/−1−ϰp1+z2ρ/2−ap/2=ϰaN−a−21+z2−a/2/−11−ϰp−1aN−a−21+z2ρ/2−ap/2+a/2+1=ϰaN−a−21+z2−a/2/−21−1+z2ρ+2/2−ap−1/2>0.This shows that(49)ηt,z=wz,t≥0,z∈ℝN,is a global solution (so local solution) to (1) with η0=w>0.Proof of Theorem 11.
(i)
Suppose thatη∈LlocΛT,N is a nontrivial local weak solution to (2) for some fixed T∈0,∞. Then, using (13) with ξ is the function defined by (18), and one obtains(50)∫ΛT,N1+z2ρ/2ηpξdzdt≤∫ΛT,NηΔξdzdt+∫ΛT,NηΔ∂ttJT1−σξdzdt+∫ℝNη0zΔ∂tJT1−σξ0,zdz+∫ℝNη1zΔJT1−σξ0,zdz.Notice that by (20) (with κ=1−σ), one has ∂tJT1−σξT,⋅≡0. Hence, by Definition 2, the choice of the test function ξ defined by (18) is allowed. Next, following the same arguments used in the proof of part (i) of Theorem 3, by the use of ε-Young inequality, one obtains
(51)∫ΔT,N1+z2ρ/2ηpξdzdt≤C∑j=15AjS,where
(52)A1S=∫ΛT,Nξ−1/p−11+z2−ρ/2p−1Δξp/p−1dzdt,A2S=∫ΛT,Nξ−1/p−11+z2−ρ/2p−1Δ∂tJT1−σξp/p−1dzdt,A3S=∫ΛT,Nξ−1/p−11+z2−ρ/2p−1Δ∂ttJT1−σξp/p−1dzdt,A4S=∫ℝNη0zΔ∂tJT1−σξ0,zdz,A5S=∫ℝNη1zΔJT1−σξ0,zdz.Notice that by (18), one has
(53)A3S=∫0Tαt−1/p−1∂ttJT1−σαtdt∫ℝNβz−1/p−11+z2−p/2p−1Δβzp/p−1dz.Hence, using Lemma6 and Lemma 9 with κ=1−σ, for sufficiently large S, one deduces that
(54)A3S≤CT−σp+1/p−1SN−2p+ρ/p−1.Furthermore, by (19) and (20) (with κ=1−σ), one has
(55)A4S=∂tJT1−σα0∫ℝNη0zΔβzdz=CT−σ∫ℝNη0zΔβzdz(56)A5S=JT1−σα0∫ℝNη1zΔβzdz=CT1−σ∫ℝNη1zΔβzdz.Next, using (36), (38), (51), (54), (55), and (56), for sufficiently large S, one obtains
(57)∫0T∫ℝN1+z2ρ/2ηpαtFz2S2mdzdt≤CT+T1−σp/p−1+T−σp+1/p−1SN−2p+ρ/p−1+∫ℝNT−ση0z+T1−ση1zΔβzdzNotice that by (26), since η0,η1∈L1ℝN, one has
(58)limS→∞∫ℝNT−ση0z+T1−ση1zΔβzdz=0.Therefore, passing to the infimum limit asS⟶∞ in (57) and using Fatou’s lemma, since 1<p<pc, it holds that
(59)∫0T∫ℝN1+z2ρ/2ηpαtdzdt=0,which contradicts the fact that η is nontrivial.
(ii)
From the proof of part (ii) of Theorem3, one deduces that the function ηt,z=wz, where w is defined by (46), is a global solution (so local solution) to (2) with η0=w>0 and η1≡0



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