Research Article | Open Access

Sebti Kerbal, Mokhtar Kirane, "Nonexistence Results for the Cauchy Problem for Nonlinear Ultraparabolic Equations", *Abstract and Applied Analysis*, vol. 2011, Article ID 149091, 10 pages, 2011. https://doi.org/10.1155/2011/149091

# Nonexistence Results for the Cauchy Problem for Nonlinear Ultraparabolic Equations

**Academic Editor:**Toka Diagana

#### 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 equation posed for , and subject to the initial conditions Here are real numbers. Then we extend our results to systems of the form for , subject to the initial conditions and where , and are real numbers. We take the nonlinearities in (1.1) and 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 [4โ7]. 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 [10โ12], 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 which corresponds to (1.1) in the absence of and with .

In his article [19] corresponding to , Fujita proved that(i), then no global positive solutions for any nonnegative initial data exist;(ii), global small data solutions exist while global solutions for large data do not exist.

The borderline case has been decided by Hayakawa [20] for and then by Kobayashi et al. [21] for any ; In case , the exponent is called the critical exponent.

For (1.5), Samarskii et al. [22] showed that the critical exponent is .

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 is called a weak solution to (1.1) if
for any test function ; and ,โโ.

Note that every weak solution is classical near the points where 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: and then by regularizing the degenerate term ; so, the regular equation is obtained. Consequently, one obtains, for small time , -uniform estimates of solutions, namely, estimates which are independent on the โparabolicityโ 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 . If , 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 is a global solution.

If we write
and estimate using the -Young inequality, we obtain
Similarly, we have
where .

Now, using (2.5) and (2.6), we obtain
If we choose , then we get the estimate
for some positive constant . Observe that the right-hand side of (2.8) is free of the unknown function .

At this stage, we introduce the smooth nonincreasing function such that
Let us take in (2.8)
with and being positive real number.

Let us now pass to the new variables
We have
Whereupon
with
where .

Now, we want to pass to the limit as in (2.13) under the constraint . We have to consider two cases.(i)Either 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 , and in this case, we get in particular
where .

Now, to conclude, we rely on the estimate
which is obtained by using the Hรถlder inequality.

Passing to the limit as in (2.16), we obtain
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 -System with a 2-Dimensional Time

In this section, we extend the analysis of the previous section to the case of a -system of 2-time equations. More precisely, we consider the system for , subject to the initial conditions and where ,โโ, and are real numbers.

To lighten the presentation, let us set Let us start with the following definition.

*Definition 2.4. *We say that is a weak solution to system (2.18) if
for any test function .

Note that every weak solution is classical near the points where and are positive.

Let us set

Theorem 2.5. *Let ,โโ,โโ,โโ, and assume that
**Then system (2.18)-(2.19) admits no global weak solution whenever
*

*Proof. *Assume that the solution is global. Using Hรถlder's inequality, we obtain
for . Similarly, we have

If we set
then, using (2.23), inequalities (2.26) and (2.27) in (2.21), we my write
so
for some positive constant .

Whereupon
Using Hรถlder's inequality, we may write
At this stage, using the scaled variables (2.11), we obtain
Hence, for , we get the estimate
Observe that, following the same lines, we can also obtain the following estimate for :

To conclude, we have to consider two cases.*Case 1. *If then
A contradiction.*Case 2. *If , 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.

#### References

- A. Kolmogoroff, โZufällige Bewegungen (zur Theorie der Brownschen Bewegung),โ
*Annals of Mathematics. Second Series*, vol. 35, no. 1, pp. 116โ117, 1934. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Cercignani,
*The Boltzmann Equation and Its Applications*, vol. 67 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 1988. - P.-L. Lions, โOn Boltzmann and Landau equations,โ
*Philosophical Transactions of the Royal Society of London. Series A*, vol. 346, no. 1679, pp. 191โ204, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Antonelli, E. Barucci, and M. E. Mancino, โAsset pricing with a forward-backward stochastic differential utility,โ
*Economics Letters*, vol. 72, no. 2, pp. 151โ157, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Chapman and T. G. Cowling,
*The Mathematical Theory of Nonuniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 3rd edition, 1990. - S. Chandresekhar, โStochastic problems in physics and astronomy,โ
*Reviews of Modern Physics*, vol. 15, pp. 1โ89, 1943. View at: Publisher Site | Google Scholar | MathSciNet - J. J. Duderstadt and W. R. Martin,
*Transport Theory*, John Wiley & Sons, New York, NY, USA, 1979. - H. Ugowski, โOn differential inequalities of parabolic type with multidimensional time,โ
*Demonstratio Mathematica*, vol. 7, pp. 113โ123, 1974. View at: Google Scholar | Zentralblatt MATH - W. Walter, โParabolic differential equations and inequalities with several time variables,โ
*Mathematische Zeitschrift*, vol. 191, no. 2, pp. 319โ323, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. P. Lavrenyuk and M. O. Olīskevich, โA mixed problem for a semilinear ultraparabolic equation in an unbounded domain,โ
*Ukraïns'kiĭ Matematichniĭ Zhurnal*, vol. 59, no. 12, pp. 1661โ1673, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Lavrenyuk and N. Protsakh, โBoundary value problem for nonlinear ultraparabolic equation in unbounded with respect to time variable domain,โ
*Tatra Mountains Mathematical Publications*, vol. 38, pp. 131โ146, 2007. View at: Google Scholar | Zentralblatt MATH - N. Protsakh, โMixed problem for degenerate nonlinear ultraparabolic equation,โ
*Tatra Mountains Mathematical Publications*, vol. 43, pp. 203โ214, 2009. View at: Google Scholar | Zentralblatt MATH - E. Lanconelli, A. Pascucci, and S. Polidoro, โLinear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance,โ in
*Nonlinear Problems in Mathematical Physics and Related Topics, II*, vol. 2 of*Int. Math. Ser. (N. Y.)*, pp. 243โ265, Kluwer/Plenum, New York, NY, USA, 2002. View at: Google Scholar | Zentralblatt MATH - A. Pascucci and S. Polidoro, โOn the Cauchy problem for a nonlinear Kolmogorov equation,โ
*SIAM Journal on Mathematical Analysis*, vol. 35, no. 3, pp. 579โ595, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Citti, A. Pascucci, and S. Polidoro, โRegularity properties of viscosity solutions of a non-Hörmander degenerate equation,โ
*Journal de Mathématiques Pures et Appliquées*, vol. 80, no. 9, pp. 901โ918, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Escobedo, J. L. Vázquez, and E. Zuazua, โEntropy solutions for diffusion-convection equations with partial diffusivity,โ
*Transactions of the American Mathematical Society*, vol. 343, no. 2, pp. 829โ842, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. S. Tersenov, โUltraparabolic equations and unsteady heat transfer,โ
*Journal of Evolution Equations*, vol. 5, no. 2, pp. 277โ289, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - O. Narayan and J. Roychowdhury, โAnalyzing oscillators using multitime PDEs,โ
*IEEE Transactions on Circuits and Systems. I*, vol. 50, no. 7, pp. 894โ903, 2003. View at: Publisher Site | Google Scholar - H. Fujita, โOn the blowing up of solutions of the Cauchy problem for ${u}_{t}=\mathrm{\Delta}u+{u}^{1+\alpha}$,โ
*Journal of the Faculty of Science. University of Tokyo. Section IA*, vol. 13, pp. 109โ124, 1966. View at: Google Scholar | Zentralblatt MATH - K. Hayakawa, โOn nonexistence of global solutions of some semilinear parabolic differential equations,โ
*Proceedings of the Japan Academy*, vol. 49, pp. 503โ505, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. Kobayashi, T. Sirao, and H. Tanaka, โOn the growing up problem for semilinear heat equations,โ
*Journal of the Mathematical Society of Japan*, vol. 29, no. 3, pp. 407โ424, 1977. View at: Publisher Site | Google Scholar - A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov,
*Blow-up in Quasilinear Parabolic Equations*, vol. 19 of*de Gruyter Expositions in Mathematics*, Walter de Gruyter, Berlin, Germany, 1995. - A. S. Kalashnikov, โSome problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,โ
*Uspekhi Matematicheskikh Nauk*, vol. 42, no. 2(254), pp. 135โ176, 1987, English translation: Russina Math. Surveys, no. 42, pp. 169–222, 1987. View at: Google Scholar | Zentralblatt MATH - G. Lu, โExistence and uniqueness of global solution for certain nonlinear degenerate parabolic systems,โ
*Acta Mathematicae Applicatae Sinica. English Series*, vol. 12, no. 3, pp. 332โ336, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH - E. Mitidieri and S. N. Pokhozhaev, โA priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities,โ
*Proceeding of the Steklov Institute of Mathematics*, vol. 234, no. 3, pp. 1โ362, 2001. View at: Google Scholar

#### Copyright

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.