Nonexistence Results for the Cauchy Problem for Nonlinear Ultraparabolic Equations
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.
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  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  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 . 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. ; see also [16, 17]. In the absence of diffusion, an interesting application is mentioned in . 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  and their complementary results by Hayakawa , Kobayashi et al. , and Samarskii et al.  concerning nonexistence results for the equation which corresponds to (1.1) in the absence of and with .
In his article  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.
Definition 2.1. A function is called a weak solution to (1.1) if for any test function ; and , .
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 “parabolicity” constants of the equation as it is clearly explained in , see also [10, 24].
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 . 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 Hölder 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 , , 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
Proof. Assume that the solution is global. Using Hölder'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 Hölder'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.
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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