Nonexistence Results for the Cauchy Problem for Nonlinear Ultraparabolic Equations

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.

1. Introduction

In this paper, we will present first nonexistence results for the two-time nonlinear equationℒ𝑢∶=𝑢_𝑡1+𝑢_𝑡2−Δ(|𝑢|^𝑚)=|𝑢|^𝑝,(1.1) posed for (𝑡₁,𝑡₂,𝑥)∈𝑄=(0,+∞)×(0,+∞)×ℝ^𝑑,𝑑∈ℕ, and subject to the initial conditions𝑢𝑡₁,0;𝑥=𝜑₁𝑡₁;𝑥,𝑢0,𝑡₂;𝑥=𝜑₂𝑡₂;𝑥.(1.2) Here 𝑝>1,𝑚>0 are real numbers. Then we extend our results to systems of the form𝑢𝑣_𝑡1𝑡1+𝑢_𝑡2−Δ(|𝑢|^𝑚)=|𝑣|^𝑝,+𝑣_𝑡2−Δ(|𝑣|^𝑛)=|𝑢|^𝑞,(1.3) for (𝑡₁,𝑡₂,𝑥)∈𝑄, subject to the initial conditions𝑣𝑢𝑡𝑡₁,0,𝑥=𝜑₁𝑡₁,𝑥,𝑢0,𝑡₂,𝑥=𝜑₂𝑡₂,𝑥₁,,0,𝑥=𝜓₁𝑡₁,𝑥,𝑣0,𝑡₂,𝑥=𝜓₂𝑡₂,𝑥,(1.4) 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 equation𝑢_𝑡−Δ(𝑢^𝑚)=|𝑢|^𝑝,𝑡>0,𝑥∈ℝ^𝑑,(1.5) 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 case 𝑝=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 𝑝_crit=1+2/𝑑 is called the critical exponent.

For (1.5), Samarskii et al. [22] showed that the critical exponent is 𝑝_crit=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 function 𝑢∈𝐿^𝑚loc(𝑄)∩𝐿^𝑝loc(𝑄) is called a weak solution to (1.1) if _𝑄|𝑢|^𝑝𝜑𝑑𝑃+_𝑆𝑢0,𝑡₂;𝑥𝜑0,𝑡₂;𝑥𝑑𝑃₂+_𝑆𝑢𝑡₁,0;𝑥𝜑𝑡₁,0;𝑥𝑑𝑃₁=−_𝑄𝑢𝜑_𝑡1𝑑𝑃−_𝑄𝑢𝜑_𝑡2𝑑𝑃−_𝑄|𝑢|^𝑚Δ𝜑𝑑𝑃(2.1) for any test function 𝜑∈𝐶^∞0(𝑄); 𝑆=ℝ₊×ℝ^𝑑,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:ℒ_𝜀𝑢=ℒ𝑢−𝜀𝐷_𝑡1𝑡1,𝜀>0,(2.2) and then by regularizing the degenerate term Δ(|𝑢|^𝑚); so, the regular equation𝑢_𝑡1+𝑢_𝑡2−Δ(min{𝑘𝑢,|𝑢|^𝑚})−𝜀𝐷_𝑡1𝑡1=|𝑢|^𝑝,𝜀>0,𝑘=1,2,…(2.3) 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 𝑢𝜑_𝑡𝑖=𝑢𝜑^1/𝑝𝜑^−1/𝑝𝜑_𝑡𝑖,𝑖=1,2,(2.4) and estimate ∫_𝑄𝑢𝜑_𝑡𝑖𝑑𝑃 using the 𝜀-Young inequality, we obtain _𝑄𝑢𝜑_𝑡𝑖𝑑𝑃≤𝜀_𝑄|𝑢|^𝑝𝜑𝑑𝑃+𝐶_𝜀_𝑄𝜑^{−1/(𝑝−1)}||𝜑_𝑡𝑖||^{𝑝/(𝑝−1)}𝑑𝑃.(2.5) Similarly, we have _𝑄|𝑢|^𝑚Δ𝜑𝑑𝑃≤𝜀_𝑄|𝑢|^𝑝𝜑𝑑𝑃+𝐶_𝜀_𝑄𝜑^{−𝑚/(𝑝−𝑚)}||Δ𝜑||^{𝑝/(𝑝−𝑚)}𝑑𝑃,(2.6) where 𝑝>𝑚.
Now, using (2.5) and (2.6), we obtain _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+𝐶_𝜀_𝑄𝜑^{−1/(𝑝−1)}||𝜑^𝑡₁||^{𝑝/(𝑝−1)}+||𝜑_𝑡2||^{𝑝/(𝑝−1)}+𝜑^{−𝑚/(𝑝−𝑚)}||Δ𝜑||^{𝑝/(𝑝−𝑚)}𝑑𝑃.(2.7) If we choose 𝜀=1/4, then we get the estimate _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)}||𝜑^𝑡₁||^{𝑝/(𝑝−1)}+||𝜑_𝑡2||^{𝑝/(𝑝−1)}+𝜑^{−𝑚/(𝑝−𝑚)}||Δ𝜑||^{𝑝/(𝑝−𝑚)}𝑑𝑃=∶ℋ(𝜑),(2.8) 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 function 𝜃∶ℝ₊→[0,1] such that 𝜃(𝑧)=10,,02≤≤𝑧𝑧.≤1,(2.9) Let us take in (2.8) 𝜑𝑡₁,𝑡₂;𝑥=𝜃^𝜆𝑅𝑡¹²+𝑡²𝑅²+|𝑥|²𝑅²,(2.10) with 𝜆>𝑚𝑎𝑥{𝑝/(𝑝−1),2𝑝/(𝑝−𝑚)} and 𝑅 being positive real number.
Let us now pass to the new variables 𝜏₁=𝑅⁻²𝑡₁,𝜏₂=𝑅⁻²𝑡₂,𝑦=𝑅⁻¹𝑥.(2.11) We have 𝜑_𝑡𝑖=𝑅⁻²𝜑_𝜏𝑖,𝑖=1,2,Δ_𝑥𝜑=𝑅⁻²Δ_𝑦𝜑.(2.12) Whereupon _𝑄|𝑢|^𝑝𝜑𝑑𝑃+2_𝑆𝑢0,𝑡₂;𝑥𝜑0,𝑡₂,𝑥𝑑𝑃₂+2_𝑆𝑢𝑡₁,0;𝑥𝜑𝑡₁,0,𝑥𝑑𝑃₁≤𝐿𝑅^{4+𝑑−2𝑝/(𝑝−1)}+𝑅^{4+𝑑−2𝑝/(𝑝−𝑚)},(2.13) with 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)}<+∞,(2.14) where Ω₁={(𝜏₁,𝜏₂,𝑦)∶1≤|𝜏₁|+|𝜏₂|+𝑦≤2}.
Now, we want to pass to the limit as 𝑅→+∞ in (2.13) under the constraint 2𝑝/(𝑝−𝑚)−4−𝑑≥0. We have to consider two cases.(i)Either 2𝑝/(𝑝−𝑚)−4−𝑑>0⇔1<𝑝<𝑚+2𝑚/(2+𝑑)=𝑝_crit and in this case, the right-hand side of (2.13) will go to zero while the left-hand side is positive. Contradiction.(ii)Or 𝑝=𝑝_crit, and in this case, we get in particular_{ℝ2+×ℝ𝑑}|𝑢|^𝑝𝜑𝑑𝑃≤𝐶⟹lim_𝑅→+∞_𝐶𝑅|𝑢|^𝑝𝜑𝑑𝑃=0,(2.15) where 𝐶_𝑅={(𝑡₁,𝑡₂;𝑥)|𝑅²≤𝑡₁+𝑡₂+|𝑥|²≤2𝑅²}.
Now, to conclude, we rely on the estimate _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),(2.16) which is obtained by using the Hölder inequality.
Passing to the limit as 𝑅→+∞ in (2.16), we obtain _{ℝ2+×ℝ𝑑}|𝑢|^𝑝𝑑𝑃+_𝑆𝑢0,𝑡₂;𝑥𝑑𝑃₂+_𝑆𝑢𝑡₁,0;𝑥𝑑𝑃₁=0.(2.17) Contradiction.

Remark 2.3. Notice that the critical exponent for the ultraparabolic equation is smaller than the one of the corresponding parabolic equation.

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

In this section, we extend the analysis of the previous section to the case of a 2×2-system of 2-time equations. More precisely, we consider the system𝑢𝑣_𝑡1𝑡1+𝑢_𝑡2−Δ(|𝑢|^𝑚)=|𝑣|^𝑝,+𝑣_𝑡2−Δ(|𝑣|^𝑛)=|𝑢|^𝑞,(2.18) for (𝑡₁,𝑡₂;𝑥)∈𝑄, subject to the initial conditions𝑣𝑢00,,𝑡𝑡₂₂,𝑥=𝜑₁𝑡₂,𝑥,𝑢𝑡₁,0,𝑥=𝜑₂𝑡₁,𝑥,,𝑥=𝜓₁𝑡₂,𝑥,𝑣𝑡₁,0,𝑥=𝜓₂𝑡₁,𝑥,(2.19) and where 0<𝑚<𝑝,  0<𝑛<𝑞, and 𝑝,𝑞>1 are real numbers.

To lighten the presentation, let us set𝐽𝐼₀₀∶=_𝑆𝑢𝑡₁,0;𝑥𝜑𝑡₁,0;𝑥𝑑𝑃₁+_𝑆𝑢0,𝑡₂;𝑥𝜑0,𝑡₂;𝑥𝑑𝑃₂,∶=_𝑆𝑣𝑡₁,0;𝑥𝜑𝑡₁,0;𝑥𝑑𝑃₁+_𝑆𝑣0,𝑡₂;𝑥𝜑0,𝑡₂;𝑥𝑑𝑃₂.(2.20) Let us start with the following definition.

Definition 2.4. We say that (𝑢,𝑣)∈(𝐿^𝑞loc(𝑄)∩𝐿^𝑚loc(𝑄))×(𝐿^𝑝loc(𝑄)∩𝐿^𝑛loc(𝑄)) is a weak solution to system (2.18) if _𝑄|𝑣|^𝑝𝜑𝑑𝑃+𝐼₀=−_𝑄𝑢𝜑_𝑡1𝑑𝑃−_𝑄𝑢𝜑_𝑡2𝑑𝑃−_𝑄|𝑢|^𝑚Δ𝜑𝑑𝑃,_𝑄|𝑢|^𝑞𝜑𝑑𝑃+𝐽₀=−_𝑄𝑣𝜑_𝑡1𝑑𝑃−_𝑄𝑣𝜑_𝑡2𝑑𝑃−_𝑄|𝑣|^𝑛Δ𝜑𝑑𝑃(2.21) for any test function 𝜑∈𝐶^∞0(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 set𝜎₁(𝑝,𝑞)=𝑞(2−(𝑑+2)𝑝)+4+𝑑𝑝𝑞−1,𝜎²(𝑝,𝑞)=𝑞(2−(𝑑+2)𝑝)+(4+𝑑)𝑚𝜎𝑝𝑞−𝑚₃,(𝑝,𝑞)=𝑞(2𝑛−(𝑑+2)𝑝)+(4+𝑑)𝑛𝑝𝑞−𝑛,𝜎⁴(𝑝,𝑞)=𝑞(2−(𝑑+2)𝑝)+(4+𝑑)𝑚𝑛𝑝𝑞−𝑚𝑛.(2.22)

Theorem 2.5. Let 𝑝>1,  𝑞>1,  𝑝>𝑛,  𝑞>𝑚, and assume that _𝑆𝑢𝑡₁,0;𝑥𝑑𝑃₁+_𝑆𝑢0,𝑡₂;𝑥𝑑𝑃₂>0,_𝑆𝑣𝑡₁,0;𝑥𝑑𝑃₁+_𝑆𝑣0,𝑡₂;𝑥𝑑𝑃₂>0.(2.23)
Then system (2.18)-(2.19) admits no global weak solution whenever max𝜎₁(𝑝,𝑞),…,𝜎₄(𝑝,𝑞),𝜎₁(𝑞,𝑝),…,𝜎₄(𝑞,𝑝)≤0.(2.24)

Proof. Assume that the solution is global. Using Hölder's inequality, we obtain _𝑄|𝑢|^𝑚||Δ𝜑||𝑑𝑃=≤_𝑄|𝑢|^𝑚𝜑^𝑚/𝑞𝜑^−𝑚/𝑞||Δ𝜑||𝑑𝑃_𝑄|𝑢|^𝑞𝜑𝑑𝑃^𝑚/𝑞_𝑄𝜑^{−𝑚/(𝑞−𝑚)}||Δ𝜑||^{𝑞/(𝑞−𝑚)}𝑑𝑃^{(𝑞−𝑚)/𝑞},(2.25)_𝑄𝑢𝜑_𝑡𝑖𝑑𝑃≤_𝑄|𝑢|^𝑞𝜑𝑑𝑃^1/𝑞_𝑄𝜑^{−1/(𝑞−1)}||𝜑_𝑡𝑖||^{(𝑞−1)/𝑞}𝑑𝑃^{(𝑞−1)/𝑞},(2.26) for 𝑖=1,2. Similarly, we have _𝑄|𝑣|^𝑛||Δ𝜑||𝑑𝑃≤_𝑄|𝑣|^𝑝𝜑𝑑𝑃^𝑛/𝑝_𝑄𝜑^{−𝑛/(𝑝−𝑛)}||Δ𝜑||^{𝑝/(𝑝−𝑛)}𝑑𝑃^{(𝑝−𝑛)/𝑝}.(2.27)
If we set 𝒜ℐ(𝑝,∶𝑛=)=_𝑄|𝑢|^𝑞_𝑄𝜑𝑑𝑃,𝒥∶=_𝑄|𝑣|^𝑝𝜑𝑑𝑃,ℬ_𝑖𝜑^{−𝑛/(𝑝−𝑛)}||Δ𝜑||^{𝑝/(𝑝−𝑛)}𝑑𝑃^{(𝑝−𝑛)/𝑝},(𝑞)=ℬ(𝑞_𝑄)𝜑=ℬ^{−1/(𝑞−1)}₁||𝜑_𝑡𝑖||^{𝑞/(𝑞−1)}𝑑𝑃^{(𝑞−1)/𝑞},(𝑞)+ℬ₂(𝑞),(2.28) then, using (2.23), inequalities (2.26) and (2.27) in (2.21), we my write 𝒥ℐ≤≤ℐ𝒥^1/𝑝_1/𝑞ℬ(𝑝)+𝒥^𝑛/𝑝𝒜(𝑝,𝑛),ℬ(𝑞)+ℐ^𝑚/𝑞𝒜(𝑞,𝑚),(2.29) so 𝒥^𝑛/𝑝≤𝐶ℐ^{𝑛/𝑝𝑞}ℬ^𝑛/𝑞(𝑞)+ℐ^{𝑚𝑛/𝑝𝑞}𝒜^𝑛/𝑝(𝑞,𝑚)(2.30) for some positive constant 𝐶.
Whereupon ℐ≤𝐶+ℐℐ^1/𝑝𝑞ℬ^1/𝑝_{𝑛/𝑝𝑞}(𝑞)ℬ(𝑝)+ℐ^{𝑚/𝑝𝑞}𝒜^1/𝑝(𝑞,𝑚)ℬ(𝑝)ℬ^𝑛/𝑝(𝑞)𝒜(𝑝,𝑛)+ℐ^{𝑚𝑛/𝑝𝑞}𝒜^𝑛/𝑝(𝑞,𝑚)𝒜(𝑝,𝑛)ℬ(𝑝).(2.31) Using Hölder's inequality, we may write ℐ≤𝐶ℬ^1/𝑝(𝑞)ℬ(𝑝)^{𝑝𝑞/(𝑝𝑞−1)}+𝒜^1/𝑝(𝑞,𝑚)ℬ(𝑝)^{𝑝𝑞/(𝑝𝑞−𝑚)}+ℬ^𝑛/𝑝(𝑞)𝒜(𝑝,𝑛)^{𝑝𝑞/(𝑝𝑞−𝑛)}+𝒜^𝑛/𝑝(𝑞,𝑚)𝒜(𝑝,𝑛)^{𝑝𝑞/(𝑝𝑞−𝑚𝑛)}.(2.32) At this stage, using the scaled variables (2.11), we obtain ℬ_𝑖𝒜(𝑝,𝑛)=𝐶𝑅^{−2+(4+𝑑)(1−𝑛/𝑝)},(𝑞)=𝐶𝑅^{−2+(4+𝑑)(1−1/𝑞)},𝑖=1,2.(2.33) Hence, for ℐ, we get the estimate ℐ≤𝐶𝑅^{−𝜎1(𝑝,𝑞)}+𝑅^{−𝜎2(𝑝,𝑞)}+𝑅^{−𝜎3(𝑝,𝑞)}+𝑅^{−𝜎4(𝑝,𝑞)}.(2.34) Observe that, following the same lines, we can also obtain the following estimate for 𝒥: 𝒥≤𝐶𝑅^{−𝜎1(𝑞,𝑝)}+𝑅^{−𝜎2(𝑞,𝑝)}+𝑅^{−𝜎3(𝑞,𝑝)}+𝑅^{−𝜎4(𝑞,𝑝)}.(2.35)
To conclude, we have to consider two cases.Case 1. If max{𝜎₁(𝑝,𝑞),…,𝜎₄(𝑝,𝑞),𝜎₁(𝑞,𝑝),…,𝜎₄(𝑞,𝑝)}<0 then lliimm_{𝑅→+∞𝑅→+∞}ℐ=_𝑄|𝑢|^𝑞𝑑𝑃=0⟹𝑢=0,𝑝.𝑝.𝒥=_𝑄|𝑣|^𝑝𝑑𝑃=0⟹𝑣=0,𝑝.𝑝.(2.36) A contradiction.Case 2. If max{𝜎₁(𝑝,𝑞),…,𝜎₄(𝑝,𝑞),𝜎₁(𝑞,𝑝),…,𝜎₄(𝑞,𝑝)}=0, we conclude following the same argument used for one equation.

Acknowledgments

The authors thank the referees for their remarks. This work is supported by Sultan Qaboos University under Grant: IG/SCI/DOMS/11/06. This work has been done during a visit of the second named author to DOMAS, Sultan Qaboos University, Muscat, Oman, which he thanks for its support and hospitality.