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∶=𝑢𝑡1+𝑢𝑡2−Δ(|𝑢|𝑚)=|𝑢|𝑝,(1.1) posed for (𝑡1,𝑡2,𝑥)∈𝑄=(0,+∞)×(0,+∞)×ℝ𝑑,𝑑∈ℕ, and subject to the initial conditions𝑢𝑡1,0;𝑥=𝜑1𝑡1;𝑥,𝑢0,𝑡2;𝑥=𝜑2𝑡2;𝑥.(1.2) Here 𝑝>1,𝑚>0 are real numbers. Then we extend our results to systems of the form𝑢𝑡1+𝑢𝑡2−Δ(|𝑢|𝑚)=|𝑣|𝑝,𝑣𝑡1+𝑣𝑡2−Δ(|𝑣|𝑛)=|𝑢|ğ‘ž,(1.3) for (𝑡1,𝑡2,𝑥)∈𝑄, subject to the initial conditions𝑢𝑡1,0,𝑥=𝜑1𝑡1,𝑥,𝑢0,𝑡2,𝑥=𝜑2𝑡2,𝑣𝑡,𝑥1,0,𝑥=𝜓1𝑡1,𝑥,𝑣0,𝑡2,𝑥=𝜓2𝑡2,,𝑥(1.4) and where 𝑝>1,ğ‘ž>1,𝑚>0, and𝑛>0 are real numbers. We take the nonlinearities |𝑢|𝑝 in (1.1) and (|𝑣|𝑝,|𝑢|ğ‘ž) 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 equation𝑢𝑡−Δ(𝑢𝑚)=|𝑢|𝑝,𝑡>0,𝑥∈ℝ𝑑,(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 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 function 𝑢∈𝐿𝑚loc(𝑄)∩𝐿𝑝loc(𝑄) is called a weak solution to (1.1) if 𝑄|𝑢|𝑝𝜑𝑑𝑃+𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2+𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1=−𝑄𝑢𝜑𝑡1𝑑𝑃−𝑄𝑢𝜑𝑡2𝑑𝑃−𝑄|𝑢|𝑚Δ𝜑𝑑𝑃(2.1) for any test function ğœ‘âˆˆğ¶âˆž0(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:ℒ𝜀𝑢=ℒ𝑢−𝜀𝐷𝑡1𝑡1,𝜀>0,(2.2) and then by regularizing the degenerate term Δ(|𝑢|𝑚); so, the regular equation𝑢𝑡1+𝑢𝑡2−Δ(min{𝑘𝑢,|𝑢|𝑚})−𝜀𝐷𝑡1𝑡1=|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 𝑢𝜑𝑡𝑖=𝑢𝜑1/𝑝𝜑−1/𝑝𝜑𝑡𝑖,𝑖=1,2,(2.4) and estimate ∫𝑄𝑢𝜑𝑡𝑖𝑑𝑃 using the 𝜀-Young inequality, we obtain 𝑄𝑢𝜑𝑡𝑖𝑑𝑃≤𝜀𝑄|𝑢|𝑝𝜑𝑑𝑃+𝐶𝜀𝑄𝜑−1/(𝑝−1)||𝜑𝑡𝑖||𝑝/(𝑝−1)𝑑𝑃.(2.5) Similarly, we have 𝑄|𝑢|𝑚Δ𝜑𝑑𝑃≤𝜀𝑄|𝑢|𝑝𝜑𝑑𝑃+𝐶𝜀𝑄𝜑−𝑚/(𝑝−𝑚)||||Δ𝜑𝑝/(𝑝−𝑚)𝑑𝑃,(2.6) where 𝑝>𝑚.
Now, using (2.5) and (2.6), we obtain 𝑄|𝑢|𝑝𝜑𝑑𝑃+𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2+𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1≤2𝜀𝑄|𝑢|𝑝𝜑𝑑𝑃+𝐶𝜀𝑄𝜑−1/(𝑝−1)||𝜑𝑡1||𝑝/(𝑝−1)+||𝜑𝑡2||𝑝/(𝑝−1)+𝜑−𝑚/(𝑝−𝑚)||||Δ𝜑𝑝/(𝑝−𝑚)𝑑𝑃.(2.7) If we choose 𝜀=1/4, then we get the estimate 𝑄|𝑢|𝑝𝜑𝑑𝑃+2𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2+2𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1≤𝐶𝑄𝜑−1/(𝑝−1)||𝜑𝑡1||𝑝/(𝑝−1)+||𝜑𝑡2||𝑝/(𝑝−1)+𝜑−𝑚/(𝑝−𝑚)||||Δ𝜑𝑝/(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)=1,0≤𝑧≤1,0,2≤𝑧.(2.9) Let us take in (2.8) 𝜑𝑡1,𝑡2;𝑥=𝜃𝜆𝑡1𝑅2+𝑡2𝑅2+|𝑥|2𝑅2,(2.10) with 𝜆>ğ‘šğ‘Žğ‘¥{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 𝜑𝑡𝑖=𝑅−2𝜑𝜏𝑖,𝑖=1,2,Δ𝑥𝜑=𝑅−2Δ𝑦𝜑.(2.12) Whereupon 𝑄|𝑢|𝑝𝜑𝑑𝑃+2𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2,𝑥𝑑𝑃2+2𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0,𝑥𝑑𝑃1𝑅≤𝐿4+𝑑−2𝑝/(𝑝−1)+𝑅4+𝑑−2𝑝/(𝑝−𝑚),(2.13) with 𝐿∶=𝐶Ω1𝜃((𝜆−1)𝑝−𝜆)/(𝑝−1)||ğœƒî…ž||𝑝/(𝑝−1)+||ğœƒî…ž||2𝑝/(𝑝−𝑚)𝜃((𝜆−2)𝑝−𝜆𝑚)/(𝑝−𝑚)+||ğœƒî…žî…ž||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)=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)Or 𝑝=𝑝crit, and in this case, we get in particularℝ2+×ℝ𝑑|𝑢|𝑝𝜑𝑑𝑃≤𝐶⟹lim𝑅→+âˆžî€œğ¶ğ‘…|𝑢|𝑝𝜑𝑑𝑃=0,(2.15) where 𝐶𝑅={(𝑡1,𝑡2;𝑥)|𝑅2≤𝑡1+𝑡2+|𝑥|2≤2𝑅2}.
Now, to conclude, we rely on the estimate 𝑄|𝑢|𝑝𝜑𝑑𝑃+𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2+𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1≤𝐶𝑅|𝑢|𝑝𝜑𝑑𝑃1/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|𝑝𝑑𝑃+𝑆𝑢0,𝑡2;𝑥𝑑𝑃2+𝑆𝑢𝑡1,0;𝑥𝑑𝑃1=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 system𝑢𝑡1+𝑢𝑡2−Δ(|𝑢|𝑚)=|𝑣|𝑝,𝑣𝑡1+𝑣𝑡2−Δ(|𝑣|𝑛)=|𝑢|ğ‘ž,(2.18) for (𝑡1,𝑡2;𝑥)∈𝑄, subject to the initial conditions𝑢0,𝑡2,𝑥=𝜑1𝑡2𝑡,𝑥,𝑢1,0,𝑥=𝜑2𝑡1,𝑣,𝑥0,𝑡2,𝑥=𝜓1𝑡2𝑡,𝑥,𝑣1,0,𝑥=𝜓2𝑡1,,𝑥(2.19) and where 0<𝑚<𝑝,  0<𝑛<ğ‘ž, and 𝑝,ğ‘ž>1 are real numbers.

To lighten the presentation, let us set𝐼0∶=𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1+𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2,𝐽0∶=𝑆𝑣𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1+𝑆𝑣0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2.(2.20) Let us start with the following definition.

Definition 2.4. We say that (𝑢,𝑣)∈(ğ¿ğ‘žloc(𝑄)∩𝐿𝑚loc(𝑄))×(𝐿𝑝loc(𝑄)∩𝐿𝑛loc(𝑄)) is a weak solution to system (2.18) if 𝑄|𝑣|𝑝𝜑𝑑𝑃+𝐼0=−𝑄𝑢𝜑𝑡1𝑑𝑃−𝑄𝑢𝜑𝑡2𝑑𝑃−𝑄|𝑢|𝑚Δ𝜑𝑑𝑃,𝑄|𝑢|ğ‘žğœ‘ğ‘‘ğ‘ƒ+𝐽0=−𝑄𝑣𝜑𝑡1𝑑𝑃−𝑄𝑣𝜑𝑡2𝑑𝑃−𝑄|𝑣|𝑛Δ𝜑𝑑𝑃(2.21) for any test function ğœ‘âˆˆğ¶âˆž0(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 setğœŽ1(𝑝,ğ‘ž)=ğ‘ž(2−(𝑑+2)𝑝)+4+ğ‘‘ğ‘ğ‘žâˆ’1,ğœŽ2(𝑝,ğ‘ž)=ğ‘ž(2−(𝑑+2)𝑝)+(4+𝑑)𝑚,ğœŽğ‘ğ‘žâˆ’ğ‘š3ğ‘ž(𝑝,ğ‘ž)=(2𝑛−(𝑑+2)𝑝)+(4+𝑑)ğ‘›ğ‘ğ‘žâˆ’ğ‘›,ğœŽ4ğ‘ž(𝑝,ğ‘ž)=(2−(𝑑+2)𝑝)+(4+𝑑)𝑚𝑛.ğ‘ğ‘žâˆ’ğ‘šğ‘›(2.22)

Theorem 2.5. Let 𝑝>1,â€‰â€‰ğ‘ž>1,  𝑝>𝑛,â€‰â€‰ğ‘ž>𝑚, and assume that 𝑆𝑢𝑡1,0;𝑥𝑑𝑃1+𝑆𝑢0,𝑡2;𝑥𝑑𝑃2>0,𝑆𝑣𝑡1,0;𝑥𝑑𝑃1+𝑆𝑣0,𝑡2;𝑥𝑑𝑃2>0.(2.23)
Then system (2.18)-(2.19) admits no global weak solution whenever î€½ğœŽmax1(𝑝,ğ‘ž),…,ğœŽ4(𝑝,ğ‘ž),ğœŽ1(ğ‘ž,𝑝),…,ğœŽ4(ğ‘ž,𝑝)≤0.(2.24)

Proof. Assume that the solution is global. Using Hölder's inequality, we obtain 𝑄|𝑢|𝑚||||Δ𝜑𝑑𝑃=𝑄|𝑢|𝑚𝜑𝑚/ğ‘žğœ‘âˆ’ğ‘š/ğ‘ž||||≤Δ𝜑𝑑𝑃𝑄|𝑢|ğ‘žî‚¶ğœ‘ğ‘‘ğ‘ƒğ‘š/ğ‘žî‚µî€œğ‘„ğœ‘âˆ’ğ‘š/(ğ‘žâˆ’ğ‘š)||||Î”ğœ‘ğ‘ž/(ğ‘žâˆ’ğ‘š)𝑑𝑃(ğ‘žâˆ’ğ‘š)/ğ‘ž,(2.25)𝑄𝑢𝜑𝑡𝑖𝑑𝑃≤𝑄|𝑢|ğ‘žî‚¶ğœ‘ğ‘‘ğ‘ƒ1/ğ‘žî‚µî€œğ‘„ğœ‘âˆ’1/(ğ‘žâˆ’1)||𝜑𝑡𝑖||(ğ‘žâˆ’1)/ğ‘žî‚¶ğ‘‘ğ‘ƒ(ğ‘žâˆ’1)/ğ‘ž,(2.26) for 𝑖=1,2. Similarly, we have 𝑄|𝑣|𝑛||||Δ𝜑𝑑𝑃≤𝑄|𝑣|𝑝𝜑𝑑𝑃𝑛/𝑝𝑄𝜑−𝑛/(𝑝−𝑛)||||Δ𝜑𝑝/(𝑝−𝑛)𝑑𝑃(𝑝−𝑛)/𝑝.(2.27)
If we set ℐ∶=𝑄|𝑢|ğ‘žî€œğœ‘ğ‘‘ğ‘ƒ,𝒥∶=𝑄|𝑣|𝑝𝜑𝑑𝑃,𝒜(𝑝,𝑛)=𝑄𝜑−𝑛/(𝑝−𝑛)||||Δ𝜑𝑝/(𝑝−𝑛)𝑑𝑃(𝑝−𝑛)/𝑝,ℬ𝑖(ğ‘ž)=𝑄𝜑−1/(ğ‘žâˆ’1)||𝜑𝑡𝑖||ğ‘ž/(ğ‘žâˆ’1)𝑑𝑃(ğ‘žâˆ’1)/ğ‘ž,ℬ(ğ‘ž)=ℬ1(ğ‘ž)+ℬ2(ğ‘ž),(2.28) then, using (2.23), inequalities (2.26) and (2.27) in (2.21), we my write ℐ≤𝒥1/𝑝ℬ(𝑝)+𝒥𝑛/𝑝𝒜(𝑝,𝑛),𝒥≤ℐ1/ğ‘žâ„¬(ğ‘ž)+ℐ𝑚/ğ‘žğ’œ(ğ‘ž,𝑚),(2.29) so 𝒥𝑛/𝑝ℐ≤𝐶𝑛/ğ‘ğ‘žâ„¬ğ‘›/ğ‘ž(ğ‘ž)+ℐ𝑚𝑛/ğ‘ğ‘žğ’œğ‘›/𝑝(ğ‘ž,𝑚)(2.30) for some positive constant 𝐶.
Whereupon ℐℐ≤𝐶1/ğ‘ğ‘žâ„¬1/𝑝(ğ‘ž)ℬ(𝑝)+ℐ𝑚/ğ‘ğ‘žğ’œ1/𝑝(ğ‘ž,𝑚)ℬ(𝑝)+ℐ𝑛/ğ‘ğ‘žâ„¬ğ‘›/𝑝(ğ‘ž)𝒜(𝑝,𝑛)+ℐ𝑚𝑛/ğ‘ğ‘žğ’œğ‘›/𝑝.(ğ‘ž,𝑚)𝒜(𝑝,𝑛)ℬ(𝑝)(2.31) Using Hölder's inequality, we may write ℬℐ≤𝐶1/𝑝(ğ‘ž)ℬ(𝑝)ğ‘ğ‘ž/(ğ‘ğ‘žâˆ’1)+𝒜1/𝑝(ğ‘ž,𝑚)ℬ(𝑝)ğ‘ğ‘ž/(ğ‘ğ‘žâˆ’ğ‘š)+ℬ𝑛/𝑝(ğ‘ž)𝒜(𝑝,𝑛)ğ‘ğ‘ž/(ğ‘ğ‘žâˆ’ğ‘›)+𝒜𝑛/𝑝(ğ‘ž,𝑚)𝒜(𝑝,𝑛)ğ‘ğ‘ž/(ğ‘ğ‘žâˆ’ğ‘šğ‘›).(2.32) At this stage, using the scaled variables (2.11), we obtain 𝒜(𝑝,𝑛)=𝐶𝑅−2+(4+𝑑)(1−𝑛/𝑝),ℬ𝑖(ğ‘ž)=𝐶𝑅−2+(4+𝑑)(1−1/ğ‘ž),𝑖=1,2.(2.33) Hence, for ℐ, we get the estimate î€½ğ‘…â„â‰¤ğ¶âˆ’ğœŽ1(𝑝,ğ‘ž)+ğ‘…âˆ’ğœŽ2(𝑝,ğ‘ž)+ğ‘…âˆ’ğœŽ3(𝑝,ğ‘ž)+ğ‘…âˆ’ğœŽ4(𝑝,ğ‘ž).(2.34) Observe that, following the same lines, we can also obtain the following estimate for 𝒥: î€½ğ‘…ğ’¥â‰¤ğ¶âˆ’ğœŽ1(ğ‘ž,𝑝)+ğ‘…âˆ’ğœŽ2(ğ‘ž,𝑝)+ğ‘…âˆ’ğœŽ3(ğ‘ž,𝑝)+ğ‘…âˆ’ğœŽ4(ğ‘ž,𝑝).(2.35)
To conclude, we have to consider two cases.Case 1. If max{ğœŽ1(𝑝,ğ‘ž),…,ğœŽ4(𝑝,ğ‘ž),ğœŽ1(ğ‘ž,𝑝),…,ğœŽ4(ğ‘ž,𝑝)}<0 then lim𝑅→+∞ℐ=𝑄|𝑢|ğ‘žğ‘‘ğ‘ƒ=0⟹𝑢=0,𝑝.𝑝.lim𝑅→+âˆžî€œğ’¥=𝑄|𝑣|𝑝𝑑𝑃=0⟹𝑣=0,𝑝.𝑝.(2.36) A contradiction.Case 2. If max{ğœŽ1(𝑝,ğ‘ž),…,ğœŽ4(𝑝,ğ‘ž),ğœŽ1(ğ‘ž,𝑝),…,ğœŽ4(ğ‘ž,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.