# Nonexistence Results for the Cauchy Problem for Nonlinear Ultraparabolic Equations

Authors: Sebti Kerbal; Mokhtar Kirane
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Nonexistence of global solutions to ultraparabolic equations and systems is presented.
Our results fill a gap in the literature on ultraparabolic equations. The
method of proof we use relies on a choice of a suitable test function in the weak
formulation of the solutions of the problems under-study.


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## Body

## 1. Introduction

In this paper, we will present first nonexistence results for the two-time nonlinear equation(1.1)Lu∶=ut1+ut2-Δ(|u|m)=|u|p,
posed for (t1,t2,x)∈Q=(0,+∞)×(0,+∞)×ℝd,d∈ℕ, and subject to the initial conditions(1.2)u(t1,0;x)=φ1(t1;x),u(0,t2;x)=φ2(t2;x).
Here p>1,m>0 are real numbers. Then we extend our results to systems of the form(1.3)ut1+ut2-Δ(|u|m)=|v|p,vt1+vt2-Δ(|v|n)=|u|q,
for (t1,t2,x)∈Q, subject to the initial conditions(1.4)u(t1,0,x)=φ1(t1,x),u(0,t2,x)=φ2(t2,x),v(t1,0,x)=ψ1(t1,x),v(0,t2,x)=ψ2(t2,x),
and where p>1,q>1,m>0, andn>0 are real numbers. We take the nonlinearities |u|p in (1.1) and (|v|p,|u|q) in (1.3) as prototypes; we could consider much more general nonlinearities.Before we present our results, let us dwell a while on the existing literature on nonlinear ultraparabolic parabolic equations known also as pluri-parabolic equations or multitime parabolic equations which we are aware of.These types of equations started in the case of linear equations with Kolmogoroff [1] in 1934; he introduced them in order to describe the probability density of a system with 2d degrees of freedom. A lot of generalizations have been made by a large number of authors since then. Nonlinear ultraparabolic equations arise in the kinetic theory of gases [2, 3]. Some stochastic processes models lead also to ultraparabolic equations [4–7]. The analysis of nonlinear ultraparabolic equations have been studied first by Ugowski [8] who studied differential inequalities of parabolic type with multidimensional time; he established, for example, a maximum principle which is very useful for applications. His results were reformulated, in a less general setting, by Walter in [9]. Many nice works on different aspects on ultraparabolic nonlinear equations have been conducted by Lavrenyuk and his collaborators [10–12], Lanconelli and his collaborators [13, 14], and Citti et al. [15]; see also [16, 17]. In the absence of diffusion, an interesting application is mentioned in [18]. Our equation and system have their applications in diffusion theory in porous media.For better positioning of our results, let us recall the pioneering results of Fujita [19] and their complementary results by Hayakawa [20], Kobayashi et al. [21], and Samarskii et al. [22] concerning nonexistence results for the equation(1.5)ut-Δ(um)=|u|p,t>0,x∈Rd,
which corresponds to (1.1) in the absence of t2 and with t1=t.In his article [19] corresponding to m=1, Fujita proved that(i)
if1<p<1+2/d, then no global positive solutions for any nonnegative initial data u0 exist;(ii)
ifp>1+2/d, global small data solutions exist while global solutions for large data do not exist.The borderline casep=1+2/d has been decided by Hayakawa [20] for d=1,2and then by Kobayashi et al. [21] for any d≥1; In case m=1, the exponent  pcrit=1+2/d  is called the critical exponent.For (1.5), Samarskii et al. [22] showed that the critical exponent is pcrit=m+2/d.The aim of this paper is to obtain the critical exponent in the sense of Fujita for (1.1) and for system (1.3). Moreover, we present critical exponents for systems of two equations.


## 2. Results

Solutions to  (1.1) subject to conditions (1.2) are meant in the following weak sense.Definition 2.1.
A functionu∈Llocm(Q)∩Llocp(Q) is called a weak solution to (1.1) if
(2.1)∫Q|u|pφdP+∫Su(0,t2;x)φ(0,t2;x)dP2+∫Su(t1,0;x)φ(t1,0;x)dP1=-∫Quφt1dP-∫Quφt2dP-∫Q|u|mΔφdP
for any test function φ∈C0∞(Q); S=ℝ+×ℝd,P=(t1,t2,x) and P1=(t1,x),  P2=(t2,x).Note that every weak solution is classical near the points(t1,t2,x) where u(t1,t2,x) is positive.Two words about the local existence of solutions are in order: as it is a rule, one regularizes (1.1) by adding first a vanishing diffusion term as follows:(2.2)Lεu=Lu-εDt1t1,ε>0,
and then by regularizing the degenerate term Δ(|u|m); so, the regular equation(2.3)ut1+ut2-Δ(min⁡⁡{ku,|u|m})-εDt1t1=|u|p,ε>0,k=1,2,…
is obtained. Consequently, one obtains, for small time t1, (ε,k)-uniform  estimates of solutions, namely, estimates which are independent on the “parabolicity” constants of the equation as it is clearly explained in [23], see also [10, 24].Our main first result is dealing with (1.1) subject to (1.2); it is given by the following theorem.Theorem 2.2.
Assume that∫Su(0,t2;x)dP2+∫Su(t1,0;x)dP1>0. If 1≤m<p≤m+2m/(2+d), then Problem (1.1)-(1.2) does not admit global weak solutions.Proof.
Our strategy of proof is to use the weak formulation of the solution with a suitable choice of the test function which we learnt from [25]. Assume u is a global solution.
If we write(2.4)uφti=uφ1/pφ-1/pφti,i=1,2,
and estimate ∫QuφtidP using the ε-Young inequality, we obtain
(2.5)∫QuφtidP≤ε∫Q|u|pφdP+Cε∫Qφ-1/(p-1)|φti|p/(p-1)dP.
Similarly, we have
(2.6)∫Q|u|mΔφdP≤ε∫Q|u|pφdP+Cε∫Qφ-m/(p-m)|Δφ|p/(p-m)dP,
where p>m.
Now, using (2.5) and (2.6), we obtain
(2.7)∫Q|u|pφdP+∫Su(0,t2;x)φ(0,t2;x)dP2+∫Su(t1,0;x)φ(t1,0;x)dP1≤2ε∫Q|u|pφdP+Cε∫Q(φ-1/(p-1)(|φt1|p/(p-1)+|φt2|p/(p-1))+φ-m/(p-m)|Δφ|p/(p-m))dP.
If we choose ε=1/4, then we get the estimate
(2.8)∫Q|u|pφdP+2∫Su(0,t2;x)φ(0,t2;x)dP2+2∫Su(t1,0;x)φ(t1,0;x)dP1≤C∫Q(φ-1/(p-1)(|φt1|p/(p-1)+|φt2|p/(p-1))+φ-m/(p-m)|Δφ|p/(p-m))dP=:H(φ),
for some positive constant C. Observe that the right-hand side of (2.8) is free of the unknown function u.
At this stage, we introduce the smooth nonincreasing functionθ:ℝ+→[0,1] such that
(2.9)θ(z)={1,0≤z≤1,0,2≤z.
Let us take in (2.8)
(2.10)φ(t1,t2;x)=θλ(t1R2+t2R2+|x|2R2),
with λ>max{p/(p-1),2p/(p-m)} and R being positive real number.
Let us now pass to the new variables(2.11)τ1=R-2t1,τ2=R-2t2,y=R-1x.
We have
(2.12)φti=R-2φτi,i=1,2,Δxφ=R-2Δyφ.
Whereupon
(2.13)∫Q|u|pφdP+2∫Su(0,t2;x)φ(0,t2,x)dP2+2∫Su(t1,0;x)φ(t1,0,x)dP1≤L(R4+d-2p/(p-1)+R4+d-2p/(p-m)),
with
(2.14)L∶=C∫Ω1(θ((λ-1)p-λ)/(p-1)|θ′|p/(p-1)+|θ′|2p/(p-m)θ((λ-2)p-λm)/(p-m)+|θ′′|p/(p-m)θ((λ-1)p-λm)/(p-m))<+∞,
where Ω1={(τ1,τ2,y):1≤|τ1|+|τ2|+y≤2}.
Now, we want to pass to the limit asR→+∞ in (2.13) under the constraint 2p/(p-m)-4-d≥0. We have to consider two cases.(i)
Either2p/(p-m)-4-d>0⇔1<p<m+2m/(2+d)=pcrit and in this case, the right-hand side of (2.13) will go to zero while the left-hand side is positive. Contradiction.(ii)
Orp=pcrit, and in this case, we get in particular(2.15)∫R+2×Rd|u|pφdP≤C⟹lim⁡R→+∞⁡∫CR|u|pφdP=0,
where CR={(t1,t2;x)|R2≤t1+t2+|x|2≤2R2}.
Now, to conclude, we rely on the estimate(2.16)∫Q|u|pφdP+∫Su(0,t2;x)φ(0,t2;x)dP2+∫Su(t1,0;x)φ(t1,0;x)dP1≤(∫CR|u|pφdP)1/pH(φ),
which is obtained by using the Hölder inequality.
Passing to the limit asR→+∞ in (2.16), we obtain
(2.17)∫R+2×Rd|u|pdP+∫Su(0,t2;x)dP2+∫Su(t1,0;x)dP1=0.
Contradiction.Remark 2.3.
Notice that the critical exponent for the ultraparabolic equation issmaller than the one of the corresponding parabolic equation.


### 2.1. The Case of a2×2-System with a 2-Dimensional Time

In this section, we extend the analysis of the previous section to the case of a2×2-system of 2-time equations. More precisely, we consider the system(2.18)ut1+ut2-Δ(|u|m)=|v|p,vt1+vt2-Δ(|v|n)=|u|q,
for (t1,t2;x)∈Q, subject to the initial conditions(2.19)u(0,t2,x)=φ1(t2,x),u(t1,0,x)=φ2(t1,x),v(0,t2,x)=ψ1(t2,x),v(t1,0,x)=ψ2(t1,x),
and where 0<m<p,  0<n<q, and p,q>1 are real numbers.To lighten the presentation, let us set(2.20)I0∶=∫Su(t1,0;x)φ(t1,0;x)dP1+∫Su(0,t2;x)φ(0,t2;x)dP2,J0∶=∫Sv(t1,0;x)φ(t1,0;x)dP1+∫Sv(0,t2;x)φ(0,t2;x)dP2.
Let us start with the following definition.Definition 2.4.
We say that(u,v)∈(Llocq(Q)∩Llocm(Q))×(Llocp(Q)∩Llocn(Q)) is a weak solution to system (2.18) if
(2.21)∫Q|v|pφdP+I0=-∫Quφt1dP-∫Quφt2dP-∫Q|u|mΔφdP,∫Q|u|qφdP+J0=-∫Qvφt1dP-∫Qvφt2dP-∫Q|v|nΔφdP
for any test function φ∈C0∞(Q).Note that every weak solution is classical near the points(t1,t2,x) where u(t1,t2,x) and v(t1,t2,x) are positive.Let us set(2.22)σ1(p,q)=q(2-(d+2)p)+4+dpq-1,σ2(p,q)=q(2-(d+2)p)+(4+d)mpq-m,σ3(p,q)=q(2n-(d+2)p)+(4+d)npq-n,σ4(p,q)=q(2-(d+2)p)+(4+d)mnpq-mn.Theorem 2.5.
Letp>1,  q>1,  p>n,  q>m, and assume that
(2.23)∫Su(t1,0;x)dP1+∫Su(0,t2;x)dP2>0,∫Sv(t1,0;x)dP1+∫Sv(0,t2;x)dP2>0.
Then system (2.18)-(2.19) admits no global weak solution whenever
(2.24)max⁡{σ1(p,q),…,σ4(p,q),σ1(q,p),…,σ4(q,p)}≤0.Proof.
Assume that the solution is global. Using Hölder's inequality, we obtain(2.25)∫Q|u|m|Δφ|dP=∫Q|u|mφm/qφ-m/q|Δφ|dP≤(∫Q|u|qφdP)m/q(∫Qφ-m/(q-m)|Δφ|q/(q-m)dP)(q-m)/q,(2.26)∫QuφtidP≤(∫Q|u|qφdP)1/q(∫Qφ-1/(q-1)|φti|(q-1)/qdP)(q-1)/q,
for i=1,2. Similarly, we have
(2.27)∫Q|v|n|Δφ|dP≤(∫Q|v|pφdP)n/p(∫Qφ-n/(p-n)|Δφ|p/(p-n)dP)(p-n)/p.
If we set(2.28)I∶=∫Q|u|qφdP,J∶=∫Q|v|pφdP,A(p,n)=(∫Qφ-n/(p-n)|Δφ|p/(p-n)dP)(p-n)/p,Bi(q)=(∫Qφ-1/(q-1)|φti|q/(q-1)dP)(q-1)/q,B(q)=B1(q)+B2(q),
then, using (2.23), inequalities (2.26) and (2.27) in (2.21), we my write
(2.29)I≤J1/pB(p)+Jn/pA(p,n),J≤I1/qB(q)+Im/qA(q,m),
so
(2.30)Jn/p≤C{In/pqBn/q(q)+Imn/pqAn/p(q,m)}
for some positive constant C.
Whereupon(2.31)I≤C{I1/pqB1/p(q)B(p)+Im/pqA1/p(q,m)B(p)+In/pqBn/p(q)A(p,n)+Imn/pqAn/p(q,m)A(p,n)B(p)}.
Using Hölder's inequality, we may write
(2.32)I≤C{(B1/p(q)B(p))pq/(pq-1)+(A1/p(q,m)B(p))pq/(pq-m)+(Bn/p(q)A(p,n))pq/(pq-n)+(An/p(q,m)A(p,n))pq/(pq-mn)}.
At this stage, using the scaled variables (2.11), we obtain
(2.33)A(p,n)=CR-2+(4+d)(1-n/p),Bi(q)=CR-2+(4+d)(1-1/q),i=1,2.
Hence, for ℐ, we get the estimate
(2.34)I≤C{R-σ1(p,q)+R-σ2(p,q)+R-σ3(p,q)+R-σ4(p,q)}.
Observe that, following the same lines, we can also obtain the following estimate for 𝒥:
(2.35)J≤C{R-σ1(q,p)+R-σ2(q,p)+R-σ3(q,p)+R-σ4(q,p)}.
To conclude, we have to consider two cases.Case 1.
Ifmax{σ1(p,q),…,σ4(p,q),σ1(q,p),…,σ4(q,p)}<0 then
(2.36)lim⁡R→+∞⁡I=∫Q|u|qdP=0⟹u=0,p.p.lim⁡R→+∞⁡J=∫Q|v|pdP=0⟹v=0,p.p.
A contradiction.Case 2.
Ifmax{σ1(p,q),…,σ4(p,q),σ1(q,p),…,σ4(q,p)}=0, we conclude following the same argument used for one equation.


## 2.1. The Case of a2×2-System with a 2-Dimensional Time

In this section, we extend the analysis of the previous section to the case of a2×2-system of 2-time equations. More precisely, we consider the system(2.18)ut1+ut2-Δ(|u|m)=|v|p,vt1+vt2-Δ(|v|n)=|u|q,
for (t1,t2;x)∈Q, subject to the initial conditions(2.19)u(0,t2,x)=φ1(t2,x),u(t1,0,x)=φ2(t1,x),v(0,t2,x)=ψ1(t2,x),v(t1,0,x)=ψ2(t1,x),
and where 0<m<p,  0<n<q, and p,q>1 are real numbers.To lighten the presentation, let us set(2.20)I0∶=∫Su(t1,0;x)φ(t1,0;x)dP1+∫Su(0,t2;x)φ(0,t2;x)dP2,J0∶=∫Sv(t1,0;x)φ(t1,0;x)dP1+∫Sv(0,t2;x)φ(0,t2;x)dP2.
Let us start with the following definition.Definition 2.4.
We say that(u,v)∈(Llocq(Q)∩Llocm(Q))×(Llocp(Q)∩Llocn(Q)) is a weak solution to system (2.18) if
(2.21)∫Q|v|pφdP+I0=-∫Quφt1dP-∫Quφt2dP-∫Q|u|mΔφdP,∫Q|u|qφdP+J0=-∫Qvφt1dP-∫Qvφt2dP-∫Q|v|nΔφdP
for any test function φ∈C0∞(Q).Note that every weak solution is classical near the points(t1,t2,x) where u(t1,t2,x) and v(t1,t2,x) are positive.Let us set(2.22)σ1(p,q)=q(2-(d+2)p)+4+dpq-1,σ2(p,q)=q(2-(d+2)p)+(4+d)mpq-m,σ3(p,q)=q(2n-(d+2)p)+(4+d)npq-n,σ4(p,q)=q(2-(d+2)p)+(4+d)mnpq-mn.Theorem 2.5.
Letp>1,  q>1,  p>n,  q>m, and assume that
(2.23)∫Su(t1,0;x)dP1+∫Su(0,t2;x)dP2>0,∫Sv(t1,0;x)dP1+∫Sv(0,t2;x)dP2>0.
Then system (2.18)-(2.19) admits no global weak solution whenever
(2.24)max⁡{σ1(p,q),…,σ4(p,q),σ1(q,p),…,σ4(q,p)}≤0.Proof.
Assume that the solution is global. Using Hölder's inequality, we obtain(2.25)∫Q|u|m|Δφ|dP=∫Q|u|mφm/qφ-m/q|Δφ|dP≤(∫Q|u|qφdP)m/q(∫Qφ-m/(q-m)|Δφ|q/(q-m)dP)(q-m)/q,(2.26)∫QuφtidP≤(∫Q|u|qφdP)1/q(∫Qφ-1/(q-1)|φti|(q-1)/qdP)(q-1)/q,
for i=1,2. Similarly, we have
(2.27)∫Q|v|n|Δφ|dP≤(∫Q|v|pφdP)n/p(∫Qφ-n/(p-n)|Δφ|p/(p-n)dP)(p-n)/p.
If we set(2.28)I∶=∫Q|u|qφdP,J∶=∫Q|v|pφdP,A(p,n)=(∫Qφ-n/(p-n)|Δφ|p/(p-n)dP)(p-n)/p,Bi(q)=(∫Qφ-1/(q-1)|φti|q/(q-1)dP)(q-1)/q,B(q)=B1(q)+B2(q),
then, using (2.23), inequalities (2.26) and (2.27) in (2.21), we my write
(2.29)I≤J1/pB(p)+Jn/pA(p,n),J≤I1/qB(q)+Im/qA(q,m),
so
(2.30)Jn/p≤C{In/pqBn/q(q)+Imn/pqAn/p(q,m)}
for some positive constant C.
Whereupon(2.31)I≤C{I1/pqB1/p(q)B(p)+Im/pqA1/p(q,m)B(p)+In/pqBn/p(q)A(p,n)+Imn/pqAn/p(q,m)A(p,n)B(p)}.
Using Hölder's inequality, we may write
(2.32)I≤C{(B1/p(q)B(p))pq/(pq-1)+(A1/p(q,m)B(p))pq/(pq-m)+(Bn/p(q)A(p,n))pq/(pq-n)+(An/p(q,m)A(p,n))pq/(pq-mn)}.
At this stage, using the scaled variables (2.11), we obtain
(2.33)A(p,n)=CR-2+(4+d)(1-n/p),Bi(q)=CR-2+(4+d)(1-1/q),i=1,2.
Hence, for ℐ, we get the estimate
(2.34)I≤C{R-σ1(p,q)+R-σ2(p,q)+R-σ3(p,q)+R-σ4(p,q)}.
Observe that, following the same lines, we can also obtain the following estimate for 𝒥:
(2.35)J≤C{R-σ1(q,p)+R-σ2(q,p)+R-σ3(q,p)+R-σ4(q,p)}.
To conclude, we have to consider two cases.Case 1.
Ifmax{σ1(p,q),…,σ4(p,q),σ1(q,p),…,σ4(q,p)}<0 then
(2.36)lim⁡R→+∞⁡I=∫Q|u|qdP=0⟹u=0,p.p.lim⁡R→+∞⁡J=∫Q|v|pdP=0⟹v=0,p.p.
A contradiction.Case 2.
Ifmax{σ1(p,q),…,σ4(p,q),σ1(q,p),…,σ4(q,p)}=0, we conclude following the same argument used for one equation.



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