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Abstract and Applied Analysis
Volume 2011, Article ID 149091, 10 pages
http://dx.doi.org/10.1155/2011/149091
Research Article

Nonexistence Results for the Cauchy Problem for Nonlinear Ultraparabolic Equations

1Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khodh, Muscat 123, Oman
2Laboratoire de Mathématiques, Image et Applications, Pôle Sciences et Technologies, Université de La Rochelle, Avenue M. Crépeau, 17042 La Rochelle, France

Received 23 March 2011; Revised 21 May 2011; Accepted 14 June 2011

Academic Editor: Toka Diagana

Copyright © 2011 Sebti Kerbal and Mokhtar Kirane. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 equation𝑢=𝑢𝑡1+𝑢𝑡2Δ(|𝑢|𝑚)=|𝑢|𝑝,(1.1) posed for (𝑡1,𝑡2,𝑥)𝑄=(0,+)×(0,+)×𝑑,𝑑, and subject to the initial conditions𝑢𝑡1,0;𝑥=𝜑1𝑡1;𝑥,𝑢0,𝑡2;𝑥=𝜑2𝑡2;𝑥.(1.2) Here 𝑝>1,𝑚>0 are real numbers. Then we extend our results to systems of the form𝑢𝑡1+𝑢𝑡2Δ(|𝑢|𝑚)=|𝑣|𝑝,𝑣𝑡1+𝑣𝑡2Δ(|𝑣|𝑛)=|𝑢|𝑞,(1.3) for (𝑡1,𝑡2,𝑥)𝑄, subject to the initial conditions𝑢𝑡1,0,𝑥=𝜑1𝑡1,𝑥,𝑢0,𝑡2,𝑥=𝜑2𝑡2,𝑣𝑡,𝑥1,0,𝑥=𝜓1𝑡1,𝑥,𝑣0,𝑡2,𝑥=𝜓2𝑡2,,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 [47]. 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 [1012], 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 equation𝑢𝑡Δ(𝑢𝑚)=|𝑢|𝑝,𝑡>0,𝑥𝑑,(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 𝑑1; 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 𝑢𝐿𝑚loc(𝑄)𝐿𝑝loc(𝑄) is called a weak solution to (1.1) if 𝑄|𝑢|𝑝𝜑𝑑𝑃+𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2+𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1=𝑄𝑢𝜑𝑡1𝑑𝑃𝑄𝑢𝜑𝑡2𝑑𝑃𝑄|𝑢|𝑚Δ𝜑𝑑𝑃(2.1) for any test function 𝜑𝐶0(𝑄); 𝑆=+×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:𝜀𝑢=𝑢𝜀𝐷𝑡1𝑡1,𝜀>0,(2.2) and then by regularizing the degenerate term Δ(|𝑢|𝑚); so, the regular equation𝑢𝑡1+𝑢𝑡2Δ(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 1𝑚<𝑝𝑚+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/𝑝𝜑1/𝑝𝜑𝑡𝑖,𝑖=1,2,(2.4) and estimate 𝑄𝑢𝜑𝑡𝑖𝑑𝑃 using the 𝜀-Young inequality, we obtain 𝑄𝑢𝜑𝑡𝑖𝑑𝑃𝜀𝑄|𝑢|𝑝𝜑𝑑𝑃+𝐶𝜀𝑄𝜑1/(𝑝1)||𝜑𝑡𝑖||𝑝/(𝑝1)𝑑𝑃.(2.5) Similarly, we have 𝑄|𝑢|𝑚Δ𝜑𝑑𝑃𝜀𝑄|𝑢|𝑝𝜑𝑑𝑃+𝐶𝜀𝑄𝜑𝑚/(𝑝𝑚)||||Δ𝜑𝑝/(𝑝𝑚)𝑑𝑃,(2.6) where 𝑝>𝑚.
Now, using (2.5) and (2.6), we obtain 𝑄|𝑢|𝑝𝜑𝑑𝑃+𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2+𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃12𝜀𝑄|𝑢|𝑝𝜑𝑑𝑃+𝐶𝜀𝑄𝜑1/(𝑝1)||𝜑𝑡1||𝑝/(𝑝1)+||𝜑𝑡2||𝑝/(𝑝1)+𝜑𝑚/(𝑝𝑚)||||Δ𝜑𝑝/(𝑝𝑚)𝑑𝑃.(2.7) If we choose 𝜀=1/4, then we get the estimate 𝑄|𝑢|𝑝𝜑𝑑𝑃+2𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2+2𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1𝐶𝑄𝜑1/(𝑝1)||𝜑𝑡1||𝑝/(𝑝1)+||𝜑𝑡2||𝑝/(𝑝1)+𝜑𝑚/(𝑝𝑚)||||Δ𝜑𝑝/(𝑝𝑚)𝑑𝑃=(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+[0,1] such that 𝜃(𝑧)=1,0𝑧1,0,2𝑧.(2.9) Let us take in (2.8) 𝜑𝑡1,𝑡2;𝑥=𝜃𝜆𝑡1𝑅2+𝑡2𝑅2+|𝑥|2𝑅2,(2.10) with 𝜆>𝑚𝑎𝑥{𝑝/(𝑝1),2𝑝/(𝑝𝑚)} and 𝑅 being positive real number.
Let us now pass to the new variables 𝜏1=𝑅2𝑡1,𝜏2=𝑅2𝑡2,𝑦=𝑅1𝑥.(2.11) We have 𝜑𝑡𝑖=𝑅2𝜑𝜏𝑖,𝑖=1,2,Δ𝑥𝜑=𝑅2Δ𝑦𝜑.(2.12) Whereupon 𝑄|𝑢|𝑝𝜑𝑑𝑃+2𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2,𝑥𝑑𝑃2+2𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0,𝑥𝑑𝑃1𝑅𝐿4+𝑑2𝑝/(𝑝1)+𝑅4+𝑑2𝑝/(𝑝𝑚),(2.13) with 𝐿=𝐶Ω1𝜃((𝜆1)𝑝𝜆)/(𝑝1)||𝜃||𝑝/(𝑝1)+||𝜃||2𝑝/(𝑝𝑚)𝜃((𝜆2)𝑝𝜆𝑚)/(𝑝𝑚)+||𝜃||𝑝/(𝑝𝑚)𝜃((𝜆1)𝑝𝜆𝑚)/(𝑝𝑚)<+,(2.14) where Ω1={(𝜏1,𝜏2,𝑦)1|𝜏1|+|𝜏2|+𝑦2}.
Now, we want to pass to the limit as 𝑅+ in (2.13) under the constraint 2𝑝/(𝑝𝑚)4𝑑0. We have to consider two cases.(i)Either 2𝑝/(𝑝𝑚)4𝑑>01<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 particular2+×𝑑|𝑢|𝑝𝜑𝑑𝑃𝐶lim𝑅+𝐶𝑅|𝑢|𝑝𝜑𝑑𝑃=0,(2.15) where 𝐶𝑅={(𝑡1,𝑡2;𝑥)|𝑅2𝑡1+𝑡2+|𝑥|22𝑅2}.
Now, to conclude, we rely on the estimate 𝑄|𝑢|𝑝𝜑𝑑𝑃+𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2+𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1𝐶𝑅|𝑢|𝑝𝜑𝑑𝑃1/𝑝(𝜑),(2.16) which is obtained by using the Hölder inequality.
Passing to the limit as 𝑅+ in (2.16), we obtain 2+×𝑑|𝑢|𝑝𝑑𝑃+𝑆𝑢0,𝑡2;𝑥𝑑𝑃2+𝑆𝑢𝑡1,0;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+𝑢𝑡2Δ(|𝑢|𝑚)=|𝑣|𝑝,𝑣𝑡1+𝑣𝑡2Δ(|𝑣|𝑛)=|𝑢|𝑞,(2.18) for (𝑡1,𝑡2;𝑥)𝑄, subject to the initial conditions𝑢0,𝑡2,𝑥=𝜑1𝑡2𝑡,𝑥,𝑢1,0,𝑥=𝜑2𝑡1,𝑣,𝑥0,𝑡2,𝑥=𝜓1𝑡2𝑡,𝑥,𝑣1,0,𝑥=𝜓2𝑡1,,𝑥(2.19) and where 0<𝑚<𝑝,  0<𝑛<𝑞, and 𝑝,𝑞>1 are real numbers.

To lighten the presentation, let us set𝐼0=𝑆𝑢𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1+𝑆𝑢0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2,𝐽0=𝑆𝑣𝑡1𝜑𝑡,0;𝑥1,0;𝑥𝑑𝑃1+𝑆𝑣0,𝑡2𝜑;𝑥0,𝑡2;𝑥𝑑𝑃2.(2.20) Let us start with the following definition.

Definition 2.4. We say that (𝑢,𝑣)(𝐿𝑞loc(𝑄)𝐿𝑚loc(𝑄))×(𝐿𝑝loc(𝑄)𝐿𝑛loc(𝑄)) is a weak solution to system (2.18) if 𝑄|𝑣|𝑝𝜑𝑑𝑃+𝐼0=𝑄𝑢𝜑𝑡1𝑑𝑃𝑄𝑢𝜑𝑡2𝑑𝑃𝑄|𝑢|𝑚Δ𝜑𝑑𝑃,𝑄|𝑢|𝑞𝜑𝑑𝑃+𝐽0=𝑄𝑣𝜑𝑡1𝑑𝑃𝑄𝑣𝜑𝑡2𝑑𝑃𝑄|𝑣|𝑛Δ𝜑𝑑𝑃(2.21) for any test function 𝜑𝐶0(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+𝑑𝑝𝑞1,𝜎2(𝑝,𝑞)=𝑞(2(𝑑+2)𝑝)+(4+𝑑)𝑚,𝜎𝑝𝑞𝑚3𝑞(𝑝,𝑞)=(2𝑛(𝑑+2)𝑝)+(4+𝑑)𝑛𝑝𝑞𝑛,𝜎4𝑞(𝑝,𝑞)=(2(𝑑+2)𝑝)+(4+𝑑)𝑚𝑛.𝑝𝑞𝑚𝑛(2.22)

Theorem 2.5. Let 𝑝>1,  𝑞>1,  𝑝>𝑛,  𝑞>𝑚, and assume that 𝑆𝑢𝑡1,0;𝑥𝑑𝑃1+𝑆𝑢0,𝑡2;𝑥𝑑𝑃2>0,𝑆𝑣𝑡1,0;𝑥𝑑𝑃1+𝑆𝑣0,𝑡2;𝑥𝑑𝑃2>0.(2.23)
Then system (2.18)-(2.19) admits no global weak solution whenever 𝜎max1(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||||Δ𝜑𝑑𝑃=𝑄|𝑢|𝑚𝜑𝑚/𝑞𝜑𝑚/𝑞||||Δ𝜑𝑑𝑃𝑄|𝑢|𝑞𝜑𝑑𝑃𝑚/𝑞𝑄𝜑𝑚/(𝑞𝑚)||||Δ𝜑𝑞/(𝑞𝑚)𝑑𝑃(𝑞𝑚)/𝑞,(2.25)𝑄𝑢𝜑𝑡𝑖𝑑𝑃𝑄|𝑢|𝑞𝜑𝑑𝑃1/𝑞𝑄𝜑1/(𝑞1)||𝜑𝑡𝑖||(𝑞1)/𝑞𝑑𝑃(𝑞1)/𝑞,(2.26) for 𝑖=1,2. Similarly, we have 𝑄|𝑣|𝑛||||Δ𝜑𝑑𝑃𝑄|𝑣|𝑝𝜑𝑑𝑃𝑛/𝑝𝑄𝜑𝑛/(𝑝𝑛)||||Δ𝜑𝑝/(𝑝𝑛)𝑑𝑃(𝑝𝑛)/𝑝.(2.27)
If we set =𝑄|𝑢|𝑞𝜑𝑑𝑃,𝒥=𝑄|𝑣|𝑝𝜑𝑑𝑃,𝒜(𝑝,𝑛)=𝑄𝜑𝑛/(𝑝𝑛)||||Δ𝜑𝑝/(𝑝𝑛)𝑑𝑃(𝑝𝑛)/𝑝,𝑖(𝑞)=𝑄𝜑1/(𝑞1)||𝜑𝑡𝑖||𝑞/(𝑞1)𝑑𝑃(𝑞1)/𝑞,(𝑞)=1(𝑞)+2(𝑞),(2.28) then, using (2.23), inequalities (2.26) and (2.27) in (2.21), we my write 𝒥1/𝑝(𝑝)+𝒥𝑛/𝑝𝒜(𝑝,𝑛),𝒥1/𝑞(𝑞)+𝑚/𝑞𝒜(𝑞,𝑚),(2.29) so 𝒥𝑛/𝑝𝐶𝑛/𝑝𝑞𝑛/𝑞(𝑞)+𝑚𝑛/𝑝𝑞𝒜𝑛/𝑝(𝑞,𝑚)(2.30) for some positive constant 𝐶.
Whereupon 𝐶1/𝑝𝑞1/𝑝(𝑞)(𝑝)+𝑚/𝑝𝑞𝒜1/𝑝(𝑞,𝑚)(𝑝)+𝑛/𝑝𝑞𝑛/𝑝(𝑞)𝒜(𝑝,𝑛)+𝑚𝑛/𝑝𝑞𝒜𝑛/𝑝.(𝑞,𝑚)𝒜(𝑝,𝑛)(𝑝)(2.31) Using Hölder's inequality, we may write 𝐶1/𝑝(𝑞)(𝑝)𝑝𝑞/(𝑝𝑞1)+𝒜1/𝑝(𝑞,𝑚)(𝑝)𝑝𝑞/(𝑝𝑞𝑚)+𝑛/𝑝(𝑞)𝒜(𝑝,𝑛)𝑝𝑞/(𝑝𝑞𝑛)+𝒜𝑛/𝑝(𝑞,𝑚)𝒜(𝑝,𝑛)𝑝𝑞/(𝑝𝑞𝑚𝑛).(2.32) At this stage, using the scaled variables (2.11), we obtain 𝒜(𝑝,𝑛)=𝐶𝑅2+(4+𝑑)(1𝑛/𝑝),𝑖(𝑞)=𝐶𝑅2+(4+𝑑)(11/𝑞),𝑖=1,2.(2.33) Hence, for , we get the estimate 𝑅𝐶𝜎1(𝑝,𝑞)+𝑅𝜎2(𝑝,𝑞)+𝑅𝜎3(𝑝,𝑞)+𝑅𝜎4(𝑝,𝑞).(2.34) Observe that, following the same lines, we can also obtain the following estimate for 𝒥: 𝑅𝒥𝐶𝜎1(𝑞,𝑝)+𝑅𝜎2(𝑞,𝑝)+𝑅𝜎3(𝑞,𝑝)+𝑅𝜎4(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 lim𝑅+=𝑄|𝑢|𝑞𝑑𝑃=0𝑢=0,𝑝.𝑝.lim𝑅+𝒥=𝑄|𝑣|𝑝𝑑𝑃=0𝑣=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.

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