On the Cauchy Problem of a Quasilinear Degenerate Parabolic Equation
By Oleinik's line method, we study the existence and the uniqueness of the classical solution of the Cauchy problem for the following equation in : , provided that is suitable small. Results of numerical experiments are reported to demonstrate that the strong solutions of the above equation may blow up in finite time.
We consider the following Cauchy problem: This problem arises in financial mathematics recently; more and more mathematicians have been interested in it. In , Antonelli et al. introduced a new model for agents' decision under risk, in which the utility function is the solution to (1.1)-(1.2); they also proved, by means of probability methods, the existence of a continuous viscosity solution of (1.1)-(1.2), which satisfies for every , , under the assumption that is uniformly Lipschitz continuous function. In , Citti et al. studied the interior regularity properties of this problem; they proved that the viscosity solutions are indeed classical solutions. On the other hand, Antonelli and Pascucci  showed that the solution found in  can be also considered as a distributional solution.
However, all the above results are obtained when is suitably small; say, the solution is local. The global weak solutions of the Cauchy problem for a more general class of equations, that contains (1.1), are obtained in [4–7], and so forth. This kind of solutions, however, is few regular and does not satisfy condition (1.3) in general.
In this paper, we will solve the Cauchy problem (1.1)-(1.2) in another simpler way and get the result as  again. Moreover, some examples are provided by numerical computation. The results of computation show that the strong solutions of the above equation may blow-up in finite time, though there exist the global weak solutions.
2. Line Method
In order to describe our method, we have to quote the well-known Prandtl system for a nonstationary boundary layer arising in an axially symmetric incompressible flow past a solid body, it has the form in a domain , where and are given functions. If we introduce the Crocco variables: we obtain the following equation for : Oleinik and Samokhin  had done excellent work in the boundary theory by the line method. Comparing this equation with (1.1), it is natural to conjecture that we are able to solve problem (1.1)-(1.2) by Qleinik's method.
Consider the following initial boundary problem: where ; its first-order derivatives and are all bounded.
Definition 2.1. A function is said to be a solution of problem (2.4)-(2.5) if has first-order derivatives in (2.4) which is continuous in , and its derivative is continuous; satisfies (2.4) in , together with condition (2.5).
Suppose that for some nonnegative number and when ,
Lemma 2.2. Let be a smooth function such that in , for . Then everywhere . Let be a smooth function such that in , for . Then everywhere in .
Proof. Let us prove the first statement of Lemma 2.2. The difference satisfies the inequality
Let . Then If we choose large enough, by the maximal principle, we know everywhere in .
Let us construct functions satisfying the conditions of Lemma 2.2. To this end, we define a twice continuously differentiable even function such that for , for , where is a function, .
When , if we chose large enough and small enough.
When , by the same reason.
Let , , . Then if we chose large enough and small enough.
Similarly, we are able to prove the second statement of Lemma 2.2.
Thus we have the following.
Let where . We will show that there exist positive constants and such that the conditions for , , imply that for .
First, we rewrite (2.6) as
Applying the operator to (2.17), then
By (2.9), (2.15), and Cauchy inequality, we are able to get where depends on and its derivatives are up to the second. Let with a positive constant to be chosen later. Then if we choose according to such that . If attains its positive maximum at , then where the constant does not depend on . At the same time, the positive maximum of in cannot be attained by maximal principle. Thus we have
So, if we let small enough such that and set then
In order to estimate the second derivatives of in , consider the function Applying the operator to both sides of (2.17), we find that
At the same time, we can calculate that and so we have By the introduced assumption that the first-order and second-order derivatives of , , and are all bounded and using Cauchy inequality, we can get from (2.29) that By the transformation , if we chose large enough, we are able to show that there exist positive constants and such that the conditions for , , imply that for . Thus we have the following.
Now let us establish uniform convergence of in . For we obtain the following equation from (2.6):
Let . Then where we have chosen small enough such that and .
If attains its positive maximal value in we can choose large enough such that and then at the maximal point we have If attains its negative minimal value in , we have Notice that at . By (2.34) and (2.35), which means that the series , whose sum has the form , is majorized by a geometrical progression and, therefore, is uniformly convergent. The fact that and its derivatives up to the second-order are bounded implies that the first derivatives of are uniformly convergent as .
It follows from (2.6) that are also uniformly convergent as .
Now, we can take ; then by the above discussion, we have the following theorem.
By the way, it is easy to prove the uniqueness of the solution for the Cauchy problem (2.4), and we omit the details here.
3. Computational Examples
In this section, a numerical simulate is made for the equations by differential method. Numerical computation of these examples shows that the strong solutions for the corresponding Cauchy problem of (1.1)-(1.2) will blow-up in finite time.
Let and , , but , . Then instead of studying the Cauchy Problem (1.1)-(1.2), we can study the following initial boundary problem: If , it is clear that if is a classical solution of (3.1), then is a strong solution of the Cauchy problem (1.1)-(1.2).
To dissect domain , suppose that and , stands for the space step-length in the axis and axis , and stands for the time step-length. Let and define . The differential scheme of the original equation is (to ensure numerical stability, here we apply arithmetic averages in order to avoid “oscillation’’ and “shifting’’ of the numerical solution)
So we get the following explicit formula:
Figure 1 shows that when , at , the numerical solutions become oscillatory; at the bifurcation of solutions occurs; when the solutions will blow-up. Similarly Figure 2 shows that when , at the bifurcation of solutions occurs; when the solutions will blow-up. Figure 3 is the spatiotemporal graphs of solutions when at (initial value) and . When , the solutions will blow-up.
Experiment 2. The initial value is unknown in the general situation; so we use random numbers as the initial value and draw graphs (see Figures 4 and 5) where changes as changes when different functions are given to .
The numerical result shows that there is a locality solution of the equation. When becomes larger, the bifurcation of solutions occurs in finite time and blow-up appears. For this problem, it is essential to have a further research.
The research was supported by the Fujian National Science Foundation of China Grant 2008J0198, 2009J1009. The authors would like to thank Professor Zhao Junning for insightful discussions and Professor Xu Chuanju for helpful suggestions and comments.
H. S. Zhan, The study of the Cauchy problem of a second order quasilinear degenerate parabolic equation and the parallelism of a Riemannian manifold, Ph.D. thesis, Xiamen University's, 2004.
O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, vol. 15 of Applied Mathematics and Mathematical Computation, Chapman & Hall/CRC, Boca Raton, Fla, USA, 1999.View at: MathSciNet