Research Article Nonexistence Results for the Cauchy Problem for Nonlinear Ultraparabolic Equations

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1 Abstract and Applied Analysis Volume 2011, Article ID , 10 pages doi: /2011/ Research Article Nonexistence Results for the Cauchy Problem for Nonlinear Ultraparabolic Equations ebti Kerbal 1 and Mokhtar Kirane 2 1 Department of Mathematics and tatistics, ultan aboos University, P.O. Box 36, Al-Khodh, Muscat 123, Oman 2 Laboratoire de Mathématiques, Image et Applications, Pôle ciences et Technologies, Université de La Rochelle, Avenue M. Crépeau, La Rochelle, France Correspondence should be addressed to ebti Kerbal, skerbal@hotmail.com Received 23 March 2011; Revised 21 May 2011; Accepted 14 June 2011 Academic Editor: Toka Diagana Copyright q Kerbal and M. 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. 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 Lu : u t1 u t2 Δ ( u m) u p, 1.1 posed for t 1,t 2,x 0, 0, R d,d N, and subject to the initial conditions u t 1, 0; x ϕ 1 t 1 ; x, u 0,t 2 ; x ϕ 2 t 2 ; x. 1.2 Here p>1, m>0 are real numbers. Then we extend our results to systems of the form u t1 u t2 Δ ( u m) v p, v t1 v t2 Δ ( v n) u q, 1.3

2 2 Abstract and Applied Analysis for t 1,t 2,x, subject to the initial conditions u t 1, 0,x ϕ 1 t 1,x, v t 1, 0,x ψ 1 t 1,x, u 0,t 2,x ϕ 2 t 2,x, v 0,t 2,x ψ 2 t 2,x, 1.4 and where p > 1, q > 1, m > 0, and n > 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. ome 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 ; seealso 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 amarskii et al. 22 concerning nonexistence results for the equation u t Δ u m u p, t > 0, x R d, 1.5 which corresponds to 1.1 in the absence of t 2 and with t 1 t. In his article 19 corresponding to m 1, Fujita proved that i if 1 <p<1 2/d, then no global positive solutions for any nonnegative initial data u 0 exist; ii if p>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, amarskii 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.

3 Abstract and Applied Analysis 3 2. Results olutions to 1.1 subject to conditions 1.2 are meant in the following weak sense. Definition 2.1. A function u L m loc Lp is called a weak solution to 1.1 if loc u p ϕdp uϕ t1 dp u 0,t 2 ; x ϕ 0,t 2 ; x dp 2 u t 1, 0; x ϕ t 1, 0; x dp 1 uϕ t2 dp u m ΔϕdP 2.1 for any test function ϕ C 0 ; R R d,p t 1,t 2,x and P 1 t 1,x, P 2 t 2,x. Note that every weak solution is classical near the points t 1,t 2,x where u t 1,t 2,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 ε u Lu εd t1 t 1, ε > 0, 2.2 and then by regularizing the degenerate term Δ u m ; so, the regular equation u t1 u t2 Δ ( min { ku, u m}) εd t1 t 1 u p, ε > 0, k 1, 2, is obtained. Consequently, one obtains, for small time t 1, 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,seealso 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 u 0,t 2; x dp 2 u t 1, 0; x dp 1 > 0.If1 m<p m 2m/ 2 d, then Problem 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/p ϕ 1/p ϕ ti, i 1, 2, 2.4 and estimate uϕ t i dp using the ε-young inequality, we obtain uϕ ti dp ε u p ϕdp C ε ϕ 1/ p 1 ϕti p/ p 1 dp. 2.5

4 4 Abstract and Applied Analysis imilarly, we have u m ΔϕdP ε u p ϕdp C ε ϕ m/ p m Δϕ p/ p m dp, 2.6 where p>m. Now, using 2.5 and 2.6, weobtain u p ϕdp u 0,t 2 ; x ϕ 0,t 2 ; x dp 2 u t 1, 0; x ϕ t 1, 0; x dp 1 ( 2ε u p ϕdp C ε 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 u p ϕdp 2 u 0,t 2 ; x ϕ 0,t 2 ; x dp 2 2 u t 1, 0; x ϕ t 1, 0; x dp 1 ( C 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 1, 0 z 1, θ z 2.9 0, 2 z. Let us take in 2.8 ( ϕ t 1,t 2 ; x θ λ t1 R t ) 2 x 2, R2 R 2 with λ>max{p/ p 1, 2p/ p m } and R being positive real number. Let us now pass to the new variables τ 1 R 2 t 1, τ 2 R 2 t 2, y R 1 x We have ϕ ti R 2 ϕ τi, i 1, 2, Δ x ϕ R 2 Δ y ϕ. 2.12

5 Abstract and Applied Analysis 5 Whereupon u p ϕdp 2 u 0,t 2 ; x ϕ 0,t 2,x dp 2 2 u t 1, 0; x ϕ t 1, 0,x dp 1 ( 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 Ω 1 { τ 1,τ 2,y :1 τ 1 τ 2 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 R 2 R d u p ϕdp C lim u p ϕdp 0, R C R 2.15 where C R { t 1,t 2 ; x R 2 t 1 t 2 x 2 2R 2 }. Now, to conclude, we rely on the estimate u p ϕdp ( C R u p ϕdp u 0,t 2 ; x ϕ 0,t 2 ; x dp 2 u t 1, 0; x ϕ t 1, 0; x dp 1 ) 1/p H ( ϕ ), 2.16 which is obtained by using the Hölder inequality. Passing to the limit as R in 2.16,weobtain u p dp R 2 R d u 0,t 2 ; x dp 2 u t 1, 0; x dp Contradiction.

6 6 Abstract and Applied Analysis Remark 2.3. Notice that the critical exponent for the ultraparabolic equation is smallerthan the one of the corresponding parabolic equation The Case of a 2 2-ystem 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 system u t1 u t2 Δ ( u m) v p, v t1 v t2 Δ ( v n) u q, 2.18 for t 1,t 2 ; x, subject to the initial conditions u 0,t 2,x ϕ 1 t 2,x, v 0,t 2,x ψ 1 t 2,x, u t 1, 0,x ϕ 2 t 1,x, v t 1, 0,x ψ 2 t 1,x, 2.19 and where 0 <m<p,0<n<q,andp, q > 1 are real numbers. To lighten the presentation, let us set I 0 : J 0 : u t 1, 0; x ϕ t 1, 0; x dp 1 v t 1, 0; x ϕ t 1, 0; x dp 1 u 0,t 2 ; x ϕ 0,t 2 ; x dp 2, v 0,t 2 ; x ϕ 0,t 2 ; x dp Let us start with the following definition. Definition 2.4. We say that u, v L q loc Lm loc Lp loc Ln is a weak solution loc to system 2.18 if v p ϕdp I 0 u q ϕdp J 0 uϕ t1 dp uϕ t2 dp u m ΔϕdP, vϕ t1 dp vϕ t2 dp v n ΔϕdP 2.21 for any test function ϕ C 0. Note that every weak solution is classical near the points t 1,t 2,x where u t 1,t 2,x and v t 1,t 2,x are positive.

7 Abstract and Applied Analysis 7 Let us set σ 1 ( p, q ) q ( 2 d 2 p ) 4 d pq 1 σ 3 ( p, q ) q ( 2n d 2 p ) 4 d n pq n ( ) q ( 2 d 2 p ) 4 d m, σ 2 p, q, pq m ( ) q ( 2 d 2 p ) 4 d mn, σ 4 p, q. pq mn 2.22 Theorem 2.5. Let p>1, q>1, p>n, q>m, and assume that u t 1, 0; x dp 1 v t 1, 0; x dp 1 u 0,t 2 ; x dp 2 > 0, v 0,t 2 ; x dp 2 > Then system admits no global weak solution whenever max { σ 1 ( p, q ),...,σ4 ( p, q ),σ1 ( q, p ),...,σ4 ( q, p )} Proof. Assume that the solution is global. Using Hölder s inequality, we obtain u m Δϕ dp u m ϕ m/q ϕ m/q Δϕ dp ( ) m/q ( ) q m /q, u q ϕdp ϕ m/ q m Δϕ q/ q m dp 2.25 ( ) 1/q ( ) q 1 /q, uϕ ti dp u q ϕdp ϕ 1/ q 1 ϕti q 1 /q dp 2.26 for i 1, 2. imilarly, we have ( ) n/p ( ) p n /p. v n Δϕ dp v p ϕdp ϕ n/ p n Δϕ p/ p n dp 2.27

8 8 Abstract and Applied Analysis If we set I : u q ϕdp, ( A ( p, n ) B i ( q ) ( J : v p ϕdp, p n /p ϕ n/ p n Δϕ dp) p/ p n, ϕ 1/ q 1 ϕti q/ q 1 dp ) q 1 /q, 2.28 B ( q ) B 1 ( q ) B2 ( q ), then, using 2.23, inequalities 2.26 and 2.27 in 2.21, we my write I J 1/p B ( p ) J n/p A ( p, n ), J I 1/q B ( q ) I m/q A ( 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 {I 1/pq B 1/p( q ) B ( p ) I m/pq A 1/p( q, m ) B ( p ) I n/pq B n/p( q ) A ( p, n ) I mn/pq A n/p( q, m ) A ( p, n ) B ( p )} 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 At this stage, using the scaled variables 2.11, weobtain A ( p, n ) CR 2 4 d 1 n/p, B i ( q ) CR 2 4 d 1 1/q, i 1,

9 Abstract and Applied Analysis 9 Hence, for I,wegettheestimate { } I C R σ1 p,q R σ2 p,q R σ3 p,q R σ 4 p,q 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 σ 4 q,p To conclude, we have to consider two cases. Case 1. If max{σ 1 p, q,...,σ 4 p, q,σ 1 q, p,...,σ 4 q, p } < 0 then lim I u q dp 0 u 0, p.p. R lim J R v p dp 0 v 0, p.p A contradiction. Case 2. If max{σ 1 p, q,...,σ 4 p, q,σ 1 q, p,...,σ 4 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 ultan aboos University under Grant: IG/CI/DOM/11/06. This work has been done during a visit of the second named author to DOMA, ultan aboos University, Muscat, Oman, which he thanks for its support and hospitality. References 1 A. Kolmogoroff, Zufällige Bewegungen zur Theorie der Brownschen Bewegung, Annals of Mathematics. econd eries, vol. 35, no. 1, pp , C. Cercignani, The Boltzmann Equation and Its Applications, vol. 67 of Applied Mathematical ciences, pringer, New York, NY, UA, P.-L. Lions, On Boltzmann and Landau equations, Philosophical Transactions of the Royal ociety of London. eries A, vol. 346, no. 1679, pp , F. Antonelli, E. Barucci, and M. E. Mancino, Asset pricing with a forward-backward stochastic differential utility, Economics Letters, vol. 72, no. 2, pp , Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and DiffusioninGases, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 3rd edition, Chandresekhar, tochastic problems in physics and astronomy, Reviews of Modern Physics, vol. 15, pp. 1 89, J. J. Duderstadt and W. R. Martin, Transport Theory, John Wiley & ons, New York, NY, UA, H. Ugowski, On differential inequalities of parabolic type with multidimensional time, Demonstratio Mathematica, vol. 7, pp , 1974.

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