On the Cauchy Problem of a Quasilinear Degenerate Parabolic Equation

Abstract

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 [0,𝑇]×𝑅²: 𝜕_𝑥𝑥𝑢+𝑢𝜕_𝑦𝑢−𝜕_𝑡𝑢=𝑓(⋅,𝑢), provided that 𝑇 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.

1. Introduction

We consider the following Cauchy problem: 𝜕_𝑥𝑥𝑢+𝑢𝜕_𝑦𝑢−𝜕_𝑡𝑢=𝑓(⋅,𝑢),(𝑡,𝑥,𝑦)∈(0,𝑇]×𝑅²,𝑢(0,⋅)=𝑢₀(𝑥,𝑦),(𝑥,𝑦)∈𝑅². (1.1)(1.2) This problem arises in financial mathematics recently; more and more mathematicians have been interested in it. In [1], 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 |𝑢(𝑥,𝑦,𝑡)−𝑢(𝜉,𝜂,𝜏)|≤𝐶_𝑇(|𝑥−𝜉|+|𝑦−𝜂|) (1.3) for every (𝑥,𝑦),(𝜉,𝜂)∈𝑅², 𝑡∈[0,𝑇), under the assumption that 𝑓 is uniformly Lipschitz continuous function. In [2], 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 [3] showed that the solution 𝑢 found in [1] can be also considered as a distributional solution.

However, all the above results are obtained when 𝑇 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 [2] 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 𝜕_𝑡𝑢+𝑢𝜕_𝑥𝑢+𝑣𝜕_𝑦𝑢=𝜕_𝑡𝑈+𝑈𝜕_𝑥𝑈+𝜕²_𝑦𝜕𝑢,_𝑥(𝑟𝑢)+𝜕_𝑦(𝑟𝑣)=0 (2.1) in a domain 𝐷={0<𝑡<𝑇,0<𝑥<𝑋,0<𝑦<∞}, where 𝑈(𝑡,𝑥) and 𝑟(𝑥) are given functions. If we introduce the Crocco variables: 𝜏=𝑡,𝜉=𝑥,𝜂=𝑢(𝑡,𝑥),𝑈(𝑡,𝑥) (2.2) we obtain the following equation for 𝑤(𝜏,𝜉,𝜂)=𝜕_𝑦𝑢/𝑈: 𝑤²𝑤_𝜂𝜂−𝑤_𝜏−𝜂𝑈𝑤_𝜉+𝐴𝑤_𝜂+𝐵𝑤=0. (2.3) Oleinik and Samokhin [8] 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: 𝑤_𝜂𝜂−𝑤_𝜏+𝑤𝑤_𝜉=𝑓(𝜏,𝜉,𝜂,𝑤),𝑤(0,𝜉,𝜂)=𝑢₀(𝜉,𝜂), (2.4)(2.5) where 𝑢₀∈𝐶²(𝑅²); its first-order derivatives and 𝑢_0𝜂𝜂 are all bounded.

Definition 2.1. A function 𝑤(𝜏,𝜉,𝜂) is said to be a solution of problem (2.4)-(2.5) if 𝑤 has first-order derivatives in (2.4) which is continuous in (0,𝑇]×𝑅², and its derivative 𝑤_𝜂𝜂 is continuous; 𝑤 satisfies (2.4) in (0,𝑇]×𝑅², together with condition (2.5).

The solution of problem (2.4)-(2.5) will be constructed as the limit of a sequence 𝑤^𝑛, 𝑛→∞, which consists of solutions of the equations 𝐿_𝑛(𝑤^𝑛)=𝑤^𝑛_𝜂𝜂−𝑤^𝑛_𝜏+𝑤^𝑛−1𝑤^𝑛_𝜉−𝑓(⋅,𝑤^𝑛𝑤)=0,_𝑛(0,𝜉,𝜂)=𝑤₀(𝜉,𝜂). (2.6)(2.7) As 𝑤⁰(𝜏,𝜉,𝜂) we take a function which is smooth in [0,𝑇]×𝑅².

Suppose that for some nonnegative number 𝑝|𝑓(⋅,𝑣)|≤𝑐(1+|𝑣|^𝑝), (2.8) and when 𝑣₁−𝑣₂≥0, 𝑐₁(𝑣₁−𝑣₂)≥𝑓(⋅,𝑣₁)−𝑓(⋅,𝑣₂)≥𝑐₂(𝑣₁−𝑣₂),max{|𝜕𝑓𝜕𝜏|,|𝜕𝑓𝜕𝜉|,|𝜕𝑓𝜕𝜕𝜂||²𝑓𝜕𝑣²|}≤𝑐. (2.9)

Lemma 2.2. Let 𝑉 be a smooth function such that 𝐿_𝑛(𝑉)≥0 in (0,𝑇)×𝑅², 𝑉≤𝑤^𝑛 for 𝜏=0. Then 𝑉≤𝑤^𝑛 everywhere (0,𝑇)×𝑅². Let 𝑉₁ be a smooth function such that 𝐿_𝑛(𝑉₁)≤0 in (0,𝑇)×𝑅², 𝑉≥𝑤^𝑛 for 𝜏=0. Then 𝑉₁≥𝑤^𝑛 everywhere in (0,𝑇)×𝑅².

Proof. Let us prove the first statement of Lemma 2.2. The difference 𝑧^𝑛=𝑤^𝑛−𝑉 satisfies the inequality 0≥𝐿_𝑛(𝑧^𝑛)=𝐿_𝑛(𝑤^𝑛)−𝐿_𝑛(𝑉)=𝑧^𝑛_𝜂𝜂−𝑧^𝑛_𝜏+𝑤^𝑛−1𝑧^𝑛_𝜉−(𝑓(⋅,𝑤^𝑛)−𝑓(⋅,𝑉)). (2.10)
Let 𝑧^𝑛₁=𝑒^−𝛼𝜏𝑧^𝑛. Then 0≥𝑧^𝑛_1𝜂𝜂−𝑧^𝑛_1𝜏+𝛼𝑧^𝑛₁+𝑤^𝑛−1𝑧^𝑛_1𝜉−𝑒^−𝛼𝜏(𝑓(⋅,𝑤^𝑛)−𝑓(⋅,𝑉))≥𝑧^𝑛_1𝜂𝜂−𝑧^𝑛_1𝜏−𝛼𝑧^𝑛₁+𝑤^𝑛−1𝑧^𝑛_1𝜉−𝑐₁𝑧^𝑛₁. (2.11) If we choose 𝛼 large enough, by the maximal principle, we know 𝑉≤𝑤^𝑛 everywhere in (0,𝑇)×𝑅².
Let us construct functions satisfying the conditions of Lemma 2.2. To this end, we define a twice continuously differentiable even function such that 𝑉₁=(1−𝑒^{−𝛽|𝜂|})𝑒^𝛽𝜏 for |𝜂|>1, 𝑉₁=𝜑(𝜂)𝑒^𝛽𝜏 for |𝜂|≤1, where 𝜑(𝜂) is a 𝐶² function, |𝜑_𝜂𝜂|≤𝑐.
When |𝜂|>1, 𝐿_𝑛(𝑉₁)=𝑉^𝑛_1𝜂𝜂−𝑉_1𝜏−𝑤^𝑛−1𝑉^𝑛_1𝜉−𝑓(⋅,𝑉₁)=−𝛽²₁𝑒^{−𝛽1|𝜂|}𝑒^𝛽𝜏−𝛽(1−𝑒^{−𝛽1|𝜂|})𝑒^𝛽𝜏−𝑓(⋅,𝑉₁)≤−𝛽²₁𝑒^{−𝛽1|𝜂|}𝑒^𝛽𝜏−𝛽(1−𝑒^{−𝛽1|𝜂|})𝑒^𝛽𝜏+𝑐(1−𝑒^{−𝛽1|𝜂|})^𝑝𝑒^𝑝𝛽𝜏+𝑐<0 (2.12) if we chose 𝛽 large enough and 𝛽𝜏≤𝑇₀ small enough.
When |𝜂|≤1, 𝐿_𝑛(𝑉₁)=𝜑_𝜂𝜂𝑒^𝛽𝜏−𝛽𝜑𝑒^𝛽𝜏−𝑓(⋅,𝑉₁)≤𝜑_𝜂𝜂𝑒^𝛽𝜏−𝛽𝜑𝑒^𝛽𝜏+𝑐(1+𝜑^𝑝𝑒^𝛽𝜏𝑝)<0 (2.13) by the same reason.
Let 𝑉=𝜓(𝜂)𝑒^−𝛼𝜏, 𝛼₁>𝜓(𝜂)≥𝛼₀>0, |𝜓_𝜂𝜂|≤𝑐. Then 𝐿_𝑛(𝑉)=𝜓_𝜂𝜂𝑒^−𝛼𝜏+𝛼𝜓𝑒^𝛼𝜏−𝑓(⋅,𝑉)≥𝜓_𝜂𝜂𝑒^−𝛼𝜏+𝛼𝜓𝑒^𝛼𝜏−𝑐(1+𝜓^𝑝𝑒^𝛼𝜏𝑝)≥0 (2.14) if we chose 𝛼 large enough and 𝛼𝜏≤𝑇₀ small enough.
Similarly, we are able to prove the second statement of Lemma 2.2.

Thus we have the following.

Lemma 2.3. Suppose that 𝑓 satisfies (2.9) and 𝑉(0,𝜉,𝜂)≤𝑤₀≤𝑉₁(0,𝜉,𝜂), then 𝑉≤𝑤^𝑛≤𝑉₁. (2.15)
The smooth functions 𝑉, 𝑉₁ can be constructed as in [8], and we omit details here.

Let Φ_𝑛=Φ=(𝑢^𝑛_𝜏)²+(𝑢^𝑛_𝜉)²+(𝑢^𝑛_𝜂)², (2.16) where 𝑢^𝑛=𝑤^𝑛. We will show that there exist positive constants 𝑀 and 𝑇 such that the conditions Φ_𝜇≤𝑀 for 𝜏≤𝑇, 𝜇≤𝑛−1, imply that Φ_𝑛≤𝑀 for 𝜏≤𝑇.

First, we rewrite (2.6) as 𝑢^𝑛_𝜂𝜂−𝑢_𝜏+𝑢^𝑛−1𝑢^𝑛_𝜉−𝑓(⋅,𝑢^𝑛)=0,(𝜏,𝜉,𝜂)∈(0,𝑇]×𝑅². (2.17)

Applying the operator 2𝑢^𝑛_𝜏(𝜕/𝜕𝜏)+2𝑢^𝑛_𝜉(𝜕/𝜕𝜉)+2𝑢^𝑛_𝜂(𝜕/𝜕𝜂) to (2.17), 2𝑢^𝑛_𝜏𝑢^𝑛_𝜏𝜂𝜂+2𝑢^𝑛_𝜏(𝑢_𝜏^𝑛−1𝑢^𝑛_𝜉+𝑢^𝑛−1𝑢^𝑛_𝜉𝜏)−2𝑢^𝑛_𝜏𝑢^𝑛_𝜏𝜏−2𝜕𝑓𝜕𝑢(𝑢^𝑛_𝜏)²−2𝑢^𝑛_𝜏𝜕𝑓,𝜕𝜏2𝑢^𝑛_𝜉𝑢^𝑛_𝜉𝜂𝜂+2𝑢^𝑛_𝜉(𝑢_𝜉^𝑛−1𝑢^𝑛_𝜉+𝑢^𝑛−1𝑢^𝑛_𝜉𝜉)−2𝑢^𝑛_𝜉𝑢^𝑛_𝜏𝜉−2𝜕𝑓𝜕𝑢(𝑢^𝑛_𝜉)²−2𝑢^𝑛_𝜉𝜕𝑓,𝜕𝜉2𝑢^𝑛_𝜂𝑢^𝑛_𝜂𝜂𝜂+2𝑢^𝑛_𝜂(𝑤_𝜂^𝑛−1𝑢^𝑛_𝜉+𝑤^𝑛−1𝑢^𝑛_𝜉𝜂)−2𝑢^𝑛_𝜂𝑢^𝑛_𝜏𝜂−2𝜕𝑓𝜕𝑢(𝑢^𝑛_𝜂)²−2𝑢^𝑛_𝜂𝜕𝑓𝜕𝜉=0, (2.18) then 𝑢^𝑛−1Φ_𝜉=(2𝑢^𝑛_𝜏𝑢^𝑛_𝜏𝜉+2𝑢^𝑛_𝜉𝑢^𝑛_𝜉𝜉+2𝑢^𝑛_𝜂𝑢^𝑛_𝜂𝜉)𝑢^𝑛−1,−Φ_𝜏=−2𝑢^𝑛_𝜏𝑢^𝑛_𝜏𝜏−2𝑢^𝑛_𝜉𝑢^𝑛_𝜉𝜏−2𝑢^𝑛_𝜂𝑢^𝑛_𝜂𝜏,Φ_𝜂𝜂=2(𝑢_𝜏𝜂)²+2𝑢^𝑛_𝜏𝑢^𝑛_𝜏𝜂𝜂+2(𝑢_𝜉𝜂)²+2𝑢^𝑛_𝜉𝑢^𝑛_𝜉𝜂𝜂+2(𝑢_𝜂𝜂)²+2𝑢^𝑛_𝜂𝑢^𝑛_𝜂𝜂𝜂,Φ_𝜂𝜂+𝑢^𝑛−1Φ_𝜉−Φ_𝜏−2𝜕𝑓𝜕𝑢Φ+2𝑢^𝑛_𝜏𝑢^𝑛_𝜉𝑢_𝜉^𝑛−1+2(𝑢^𝑛_𝜉)²𝑢_𝜉^𝑛−1+2𝑢^𝑛_𝜂𝑢^𝑛_𝜉𝑢_𝜂^𝑛−1−2𝑢^𝑛_𝜏𝜕𝑓𝜕𝜏−2𝑢^𝑛_𝜉𝜕𝑓𝜕𝜉−2𝑢^𝑛_𝜂𝜕𝑓𝜕𝜉=0. (2.19)

By (2.9), (2.15), and Cauchy inequality, we are able to get Φ_𝜂𝜂+𝑢^𝑛−1Φ_𝜉−Φ_𝜏+𝑅^𝑛Φ≥0, (2.20) where 𝑅^𝑛 depends on 𝑢^𝑛−1 and its derivatives are up to the second. Let Φ₁=Φ𝑒^−𝛾𝜏 with a positive constant 𝛾 to be chosen later. Then Φ_1𝜂𝜂+𝑢^𝑛−1Φ_1𝜉−Φ_1𝜏+(𝑅^𝑛−𝛾)Φ≥0 (2.21) if we choose 𝛾 according to 𝑀 such that 𝑅^𝑛−𝛾≤−1. If Φ₁ attains its positive maximum at 𝜏=0, then Φ₁|_𝜏=0=Φ𝑒^−𝛾𝜏|_𝜏=0=Φ|_𝜏=0≤𝑐, (2.22) where the constant 𝑐 does not depend on 𝑛. At the same time, the positive maximum of Φ₁ in (0,𝑇]×𝑅² cannot be attained by maximal principle. Thus we have Φ₁≤𝑐. (2.23)

So, if we let 𝑇₁≤𝑇 small enough such that 𝑒^𝛾𝑇1=2 and set 𝑀=2𝑐, then Φ≤𝑐𝑒^𝛾𝑇1=𝑀. (2.24)

In order to estimate the second derivatives of 𝑢^𝑛 in [0,𝑇₁]×𝑅², consider the function 𝐹=(𝑢^𝑛_𝜏𝜏)²+(𝑢^𝑛_𝜉𝜉)²+(𝑢^𝑛_𝜂𝜂)²+(𝑢^𝑛_𝜏𝜉)²+(𝑢^𝑛_𝜉𝜂)²+(𝑢^𝑛_𝜏𝜂)². (2.25) Applying the operator 𝑃=2𝑢^𝑛_𝜏𝜏𝜕²𝜕𝜏²+2𝑢^𝑛_𝜉𝜉𝜕²𝜕𝜉²+2𝑢^𝑛_𝜂𝜂𝜕²𝜕𝜂²+2𝑢^𝑛_𝜏𝜉𝜕²𝜕𝜏𝜕𝜉+2𝑢^𝑛_𝜏𝜂𝜕²𝜕𝜏𝜕𝜂+2𝑢^𝑛_𝜉𝜂𝜕²𝜕𝜉𝜕𝜂 (2.26) to both sides of (2.17), we find that 0=2𝑢^𝑛_𝜏𝜏𝑢^𝑛_{𝜂𝜂𝜏𝜏}+2𝑢^𝑛_𝜏𝜏(𝑢^𝑛−1_𝜏𝜏𝑢^𝑛_𝜉+2𝑢_𝜏^𝑛−1𝑢^𝑛_𝜉𝜏+𝑢^𝑛−1𝑢^𝑛_𝜉𝜏𝜏)−2𝑢^𝑛_𝜏𝜏𝑢^𝑛_𝜏𝜏𝜏+−2𝑢^𝑛_𝜏𝜏(𝜕²𝑓𝜕𝑢²(𝑢^𝑛_𝜏)²+𝜕𝑓𝑢𝜕𝑢^𝑛𝜏𝜏)+2𝑢^𝑛_𝜉𝜉𝑢^𝑛_{𝜂𝜂𝜉𝜉}+2𝑢^𝑛_𝜉𝜉(𝑢^𝑛−1_𝜉𝜉𝑢^𝑛_𝜉+2𝑢_𝜉^𝑛−1𝑢^𝑛_𝜉𝜉+𝑢^𝑛−1𝑢^𝑛_𝜉𝜉𝜉)−2𝑢^𝑛_𝜉𝜉𝑢^𝑛_𝜏𝜉𝜉−2𝑢^𝑛_𝜉𝜉(𝜕²𝑓𝜕𝑢²(𝑢^𝑛_𝜉)²+𝜕𝑓𝑢𝜕𝑢^𝑛𝜉𝜉)+2𝑢^𝑛_𝜂𝜂𝑢^𝑛_{𝜂𝜂𝜂𝜂}+2𝑢^𝑛_𝜂𝜂(𝑢^𝑛−1_𝜂𝜂𝑢^𝑛_𝜉+2𝑢_𝜂^𝑛−1𝑢^𝑛_𝜉𝜂+𝑢^𝑛−1𝑢^𝑛_𝜉𝜂𝜂)−2𝑢^𝑛_𝜂𝜂𝑢^𝑛_𝜏𝜉𝜉−2𝑢^𝑛_𝜂𝜂(𝜕²𝑓𝜕𝑢²(𝑢^𝑛_𝜂)²−𝜕𝑓𝑢𝜕𝑢^𝑛𝜂𝜂)+2𝑢^𝑛_𝜏𝜉𝑢^𝑛_{𝜂𝜂𝜏𝜉}+2𝑢^𝑛_𝜏𝜉(𝑢^𝑛−1_𝜏𝜉𝑢^𝑛_𝜉+𝑢_𝜏^𝑛−1𝑢^𝑛_𝜉𝜉+𝑢_𝜉^𝑛−1𝑢^𝑛_𝜉𝜏+𝑢^𝑛−1𝑢^𝑛_𝜉𝜉𝜏)−2𝑢^𝑛_𝜏𝜉𝑢^𝑛_𝜏𝜏𝜉−2𝑢^𝑛_𝜏𝜉(𝜕²𝑓𝜕𝑢²𝑢^𝑛_𝜏𝑢^𝑛_𝜉+𝜕𝑓𝑢𝜕𝑢^𝑛𝜏𝜉)+2𝑢^𝑛_𝜉𝜂𝑢^𝑛_{𝜂𝜂𝜉𝜂}+2𝑢^𝑛_𝜉𝜂(𝑢^𝑛−1_𝜉𝜂𝑢^𝑛_𝜉+𝑢_𝜉^𝑛−1𝑢^𝑛_𝜉𝜂+𝑢_𝜂^𝑛−1𝑢^𝑛_𝜉𝜉+𝑢^𝑛−1𝑢^𝑛_𝜉𝜉𝜂)−2𝑢^𝑛_𝜉𝜂𝑢^𝑛_𝜏𝜏𝜉−2𝑢^𝑛_𝜉𝜂(𝜕²𝑓𝜕𝑢²𝑢^𝑛_𝜂𝑢^𝑛_𝜉+𝜕𝑓𝑢𝜕𝑢^𝑛𝜉𝜂)+2𝑢^𝑛_𝜏𝜂𝑢^𝑛_{𝜂𝜂𝜏𝜂}+2𝑢^𝑛_𝜏𝜂(𝑢^𝑛−1_𝜏𝜂𝑢^𝑛_𝜉+𝑢_𝜏^𝑛−1𝑢^𝑛_𝜉𝜂+𝑢_𝜂^𝑛−1𝑢^𝑛_𝜉𝜏+𝑢^𝑛−1𝑢^𝑛_𝜉𝜏𝜂)−2𝑢^𝑛_𝜏𝜂𝑢^𝑛_𝜏𝜏𝜂−2𝑢^𝑛_𝜏𝜂(𝜕²𝑓𝜕𝑢²𝑢^𝑛_𝜂𝑢^𝑛_𝜏+𝜕𝑓𝑢𝜕𝑢^𝑛𝜏𝜂). (2.27)

At the same time, we can calculate that 𝐹_𝜂=2𝑢^𝑛_𝜏𝜏𝑢^𝑛_𝜏𝜏𝜂+2𝑢^𝑛_𝜉𝜉𝑢^𝑛_𝜉𝜉𝜂+2𝑢^𝑛_𝜂𝜂𝑢^𝑛_𝜂𝜂𝜂+2𝑢^𝑛_𝜏𝜉𝑢^𝑛_𝜏𝜉𝜂+2𝑢^𝑛_𝜉𝜂𝑢^𝑛_𝜉𝜂𝜂+2𝑢^𝑛_𝜏𝜂𝑢^𝑛_𝜏𝜂𝜂,𝐹_𝜂𝜂=2(𝑢^𝑛_𝜏𝜏𝜂)²+2𝑢^𝑛_𝜏𝜏𝑢^𝑛_{𝜏𝜏𝜂𝜂}+2(𝑢^𝑛_𝜉𝜉𝜂)²+2𝑢^𝑛_𝜉𝜉𝑢^𝑛_{𝜂𝜂𝜉𝜉}+2(𝑢^𝑛_𝜂𝜂𝜂)²+2𝑢^𝑛_𝜂𝜂𝑢^𝑛_{𝜂𝜂𝜂𝜂}+2(𝑢^𝑛_𝜏𝜉𝜂)²+2𝑢^𝑛_𝜏𝜉𝑢^𝑛_{𝜏𝜉𝜂𝜂}+2(𝑢^𝑛_𝜂𝜉𝜉)²+2𝑢^𝑛_𝜉𝜂𝑢^𝑛_{𝜉𝜂𝜂𝜂}+2(𝑢^𝑛_𝜏𝜂𝜂)²+2𝑢^𝑛_𝜏𝜂𝑢^𝑛_{𝜏𝜂𝜂𝜂},𝑢^𝑛−1𝐹_𝜉=𝑢^𝑛−1(2𝑢^𝑛_𝜏𝜏𝑢^𝑛_𝜏𝜏𝜉+2𝑢^𝑛_𝜉𝜉𝑢^𝑛_𝜉𝜉𝜉+2𝑢^𝑛_𝜂𝜂𝑢^𝑛_𝜂𝜂𝜉+2𝑢^𝑛_𝜏𝜉𝑢^𝑛_𝜏𝜉𝜉+2𝑢^𝑛_𝜉𝜂𝑢^𝑛_𝜉𝜂𝜉+2𝑢^𝑛_𝜏𝜂𝑢^𝑛_𝜏𝜂𝜉),−𝐹_𝜏=−(2𝑢^𝑛_𝜏𝜏𝑢^𝑛_𝜏𝜏𝜏+2𝑢^𝑛_𝜉𝜉𝑢^𝑛_𝜉𝜉𝜏+2𝑢^𝑛_𝜂𝜂𝑢^𝑛_𝜂𝜂𝜏+2𝑢^𝑛_𝜏𝜉𝑢^𝑛_𝜏𝜉𝜏+2𝑢^𝑛_𝜉𝜂𝑢^𝑛_𝜉𝜂𝜏+2𝑢^𝑛_𝜏𝜂𝑢^𝑛_𝜏𝜂𝜏), (2.28) and so we have 𝐹_𝜂𝜂+𝑢^𝑛−1𝐹_𝜉−𝐹_𝜏−2𝜕𝑓𝜕𝑢𝐹−2𝑢^𝑛_𝜏𝜏(𝑤^𝑛−1_𝜏𝜏𝑢^𝑛_𝜉+2𝑤_𝜏^𝑛−1𝑢^𝑛_𝜉𝜏)−2𝑢^𝑛_𝜉𝜉(𝑢^𝑛−1_𝜉𝜉𝑢^𝑛_𝜉+2𝑢_𝜉^𝑛−1𝑢^𝑛_𝜉𝜉)−2𝑢^𝑛_𝜂𝜂(𝑢^𝑛−1_𝜂𝜂𝑢^𝑛_𝜉+2𝑤_𝜂^𝑛−1𝑢^𝑛_𝜉𝜂)−2𝑢^𝑛_𝜏𝜉(𝑢^𝑛−1_𝜏𝜉𝑢^𝑛_𝜉+𝑢_𝜏^𝑛−1𝑢^𝑛_𝜉𝜉+𝑢_𝜉^𝑛−1𝑢^𝑛_𝜉𝜏)−2𝑢^𝑛_𝜉𝜂(𝑢^𝑛−1_𝜉𝜂𝑢^𝑛_𝜉+𝑢_𝜉^𝑛−1𝑢^𝑛_𝜉𝜂+𝑢_𝜂^𝑛−1𝑢^𝑛_𝜉𝜉)−2𝑢^𝑛_𝜏𝜂(𝑢^𝑛−1_𝜏𝜂𝑢^𝑛_𝜉+𝑢_𝜏^𝑛−1𝑢^𝑛_𝜉𝜂+𝑢_𝜂^𝑛−1𝑢^𝑛_𝜉𝜏)−2𝑢^𝑛_𝜏𝜏𝜕²𝑓𝜕𝑢²(𝑢^𝑛_𝜏)²−2𝑢^𝑛_𝜉𝜉𝜕²𝑓𝜕𝑢²(𝑢^𝑛_𝜉)²−2𝑢^𝑛_𝜂𝜂𝜕²𝑓𝜕𝑢²(𝑢^𝑛_𝜂)²−2𝑢^𝑛_𝜏𝜉𝜕²𝑓𝜕𝑢²𝑢^𝑛_𝜏𝑢^𝑛_𝜉−2𝑢^𝑛_𝜉𝜂𝜕²𝑓𝜕𝑢²𝑢^𝑛_𝜂𝑢^𝑛_𝜉−2𝑢^𝑛_𝜏𝜂𝜕²𝑓𝜕𝑢²𝑢^𝑛_𝜂𝑢^𝑛_𝜏−2𝜕𝑓𝜕𝑢[(𝑢^𝑛_𝜏)²+(𝑢^𝑛_𝜉)²+(𝑢^𝑛_𝜂)²]=0. (2.29) By the introduced assumption that the first-order and second-order derivatives of 𝑢^𝑛−1, 𝜕𝑓/𝜕𝑢, and 𝜕²𝑓/𝜕𝑢² are all bounded and using Cauchy inequality, we can get from (2.29) that 𝐹_𝜂𝜂−2𝛼𝐹_𝜂−𝑢^𝑛−1𝐹_𝜉−𝐹_𝜏+𝑅^𝑛₁𝐹≥0. (2.30) By the transformation 𝐹₁=𝐹𝑒^−𝛾𝜏, if we chose 𝛾 large enough, we are able to show that there exist positive constants 𝑀 and 𝑇 such that the conditions 𝐹_𝜇≤𝑀 for 𝜏≤𝑇, 𝜇≤𝑛−1, imply that 𝐹_𝑛≤𝑀 for 𝜏≤𝑇. Thus we have the following.

Theorem 2.4. Let 𝑤^𝑛 be the solutions of problems (2.4)-(2.5), then the derivatives of 𝑤^𝑛 up to the second-order are uniformly bounded with respect to 𝑛 in the domain (0,𝑇]×𝑅² with a small positive number 𝑇.

Now let us establish uniform convergence of 𝑤^𝑛=𝑢^𝑛 in [0,𝑇]×𝑅². For 𝑣^𝑛=𝑤^𝑛−𝑤^𝑛−1 we obtain the following equation from (2.6): 𝑣^𝑛_𝜂𝜂−𝑣^𝑛_𝜏+𝑤^𝑛−1𝑣^𝑛_𝜉−𝑣^𝑛−1𝑤_𝜉^𝑛−1−(𝑓(⋅,𝑤^𝑛)−𝑓(⋅,𝑤^𝑛−1𝑣))=0,_𝑛(0,𝜉,𝜂)=0. (2.31)

Let 𝑣^𝑛=𝑒^𝛼𝜏𝑣^𝜂₁. Then 𝑣^𝑛_1𝜂𝜂−𝑣^𝑛_1𝜏+𝑤^𝑛−1𝑣^𝑛_1𝜉−𝑣₁^𝑛−1𝑤_𝜉^𝑛−1−𝛼𝑣^𝑛₁−𝑒^−𝛼𝜏(𝑓(⋅,𝑤^𝑛)−𝑓(⋅,𝑤^𝑛−1𝑣))=0,_{𝑛1𝜂𝜂}−𝑣^𝑛_1𝜏+𝑤^𝑛−1𝑣^𝑛_1𝜉−𝑣₁^𝑛−1𝑤_𝜉^𝑛−1=𝛼𝑣^𝑛₁+𝑒^−𝛼𝜏(𝑓(⋅,𝑤^𝑛)−𝑓(⋅,𝑤^𝑛−1))=𝛼𝑣^𝑛₁+𝑒^−𝛼𝜏𝜕𝑓𝑣𝜕𝑤^𝑛1≥(𝛼−𝑐)𝑣^𝑛₁, (2.32) where we have chosen 𝜏≤𝑇 small enough such that 𝑒^−𝛼𝜏=2, and 2(𝜕𝑓/𝜕𝑤)≥−𝑐.

If 𝑣₁ attains its positive maximal value in (0,𝑇]×𝑅², we can choose 𝛼 large enough such that |𝑤_𝜉^𝑛−1𝛼−𝑐|<1, (2.33) and then at the maximal point we have (𝛼−𝑐)𝑣^𝑛₁≤−𝑣₁^𝑛−1𝑤_𝜉^𝑛−1. (2.34) If 𝑣^𝑛₁ attains its negative minimal value in (0,𝑇]×𝑅², we have (𝛼−𝑐)(−𝑣^𝑛₁)≤−𝑣₁^𝑛−1𝑤_𝜉^𝑛−1. (2.35) Notice that at 𝜏=0,𝑣^𝑛₁=𝑣^𝑛=0. By (2.34) and (2.35), max|𝑣^𝑛₁|≤𝑞max|𝑣₁^𝑛−1|,𝑞<1, (2.36) which means that the series 𝑣¹₁+𝑣²₁+⋯+𝑣^𝑛₁+⋯, whose sum has the form 𝑤^𝑛𝑒^−𝛼𝜏, is majorized by a geometrical progression and, therefore, is uniformly convergent. The fact that 𝑤^𝑛 and its derivatives up to the second-order are bounded implies that the first derivatives of 𝑤^𝑛 are uniformly convergent as 𝑛→∞.

It follows from (2.6) that 𝑤^𝑛_𝜂𝜂 are also uniformly convergent as 𝑛→∞.

Now, we can take 𝑤⁻¹=𝑤⁰=𝑤₀; then by the above discussion, we have the following theorem.

Theorem 2.5. Suppose that 𝑉(0,𝜉,𝜂)≤𝑤₀≤𝑉₁(0,𝜉,𝜂) and 𝑓 satisfies (2.9) and is suitable smooth, then there exists a small positive number 𝑇 such that the Cauchy problem (2.4) has a classical solution.

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 Ω=[0,𝐿_𝑥]×[0,𝐿_𝑦] and 𝑢(𝑥,𝑦,0)=𝑢₀(𝑥,𝑦), (𝑥,𝑦)∈Ω, but 𝑢(𝑥,𝑦,0)=0, (𝑥,𝑦)∈𝑅²/Ω. Then instead of studying the Cauchy Problem (1.1)-(1.2), we can study the following initial boundary problem: 𝜕_𝑥𝑥𝑢+𝑢𝜕_𝑦𝑢−𝜕_𝑡𝑢=𝑓(⋅,𝑢),(𝑥,𝑦,𝑡)∈Ω×(0,𝑇],𝑢(𝑥,𝑦,0)=𝑢₀(𝑥,𝑦),(𝑥,𝑦)∈Ω,𝑢|_𝜕Ω=0. (3.1) If 𝑓(⋅,0)=0, it is clear that if 𝑢(𝑥,𝑦,𝑡) is a classical solution of (3.1), then 𝑢(𝑥,𝑦,𝑡) is a strong solution of the Cauchy problem (1.1)-(1.2).

To dissect domain Ω, suppose that 𝐿_𝑥=𝐿_𝑦=2𝜋 and ℎ_𝑥=2𝜋/𝑁, ℎ_𝑦=2𝜋/𝑀 stands for the space step-length in the axis 𝑥 and axis 𝑦, and 𝑘=𝑇/𝐽 stands for the time step-length. Let Ω_ℎ={(𝑖ℎ_𝑥,𝑗ℎ_𝑦)∣0≤𝑖≤𝑁;0≤𝑗≤𝑀} and define 𝑢^𝑛_𝑖𝑗=𝑢(𝑖ℎ_𝑥,𝑗ℎ_𝑦,𝑛𝑘). 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) 𝑢^𝑛_𝑖+1,𝑗−2𝑢^𝑛_𝑖,𝑗+𝑢^𝑛_{𝑖−1,𝑗}ℎ²_𝑥+𝑢^𝑛_𝑖+1,𝑗+𝑢^𝑛_𝑖,𝑗+𝑢^𝑛_{𝑖−1,𝑗}+𝑢^𝑛_𝑖,𝑗+1+𝑢^𝑛_𝑖,𝑗+𝑢^𝑛_{𝑖,𝑗−1}6𝑢^𝑛_𝑖,𝑗+1−𝑢^𝑛_{𝑖,𝑗−1}2ℎ_𝑦−𝑢^𝑛+1_𝑖,𝑗−(1/4)(𝑢^𝑛_{𝑖−1,𝑗}+𝑢^𝑛_𝑖+1,𝑗+𝑢^𝑛_𝑖,𝑗+1+𝑢^𝑛_{𝑖,𝑗−1})𝑘=𝑓(𝑖ℎ_𝑥,𝑗ℎ_𝑦𝑢,𝑛𝑘,^{𝑛𝑖+1,𝑗}+𝑢^𝑛_𝑖,𝑗+𝑢^𝑛_{𝑖−1,𝑗}+𝑢^𝑛_𝑖,𝑗+1+𝑢^𝑛_𝑖,𝑗+𝑢^𝑛_{𝑖,𝑗−1}6𝑢),_𝑛|_𝜕Ωℎ=0,(𝑛=1,2,…),𝑢⁰_𝑖,𝑗=𝑢₀(𝑖ℎ_𝑥,𝑗ℎ_𝑦). (3.2)

So we get the following explicit formula: 𝑢^𝑛+1_𝑖,𝑗=14(𝑢^𝑛_{𝑖−1,𝑗}+𝑢^𝑛_𝑖+1,𝑗+𝑢^𝑛_𝑖,𝑗+1+𝑢^𝑛_{𝑖,𝑗−1}𝑘)+ℎ²_𝑥(𝑢^𝑛_𝑖+1,𝑗−2𝑢^𝑛_𝑖,𝑗+𝑢^𝑛_{𝑖−1,𝑗})+𝑘12ℎ_𝑦(𝑢^𝑛_𝑖+1,𝑗+2𝑢^𝑛_𝑖,𝑗+𝑢^𝑛_{𝑖−1,𝑗}+𝑢^𝑛_𝑖,𝑗+1+𝑢^𝑛_{𝑖,𝑗−1})(𝑢^𝑛_𝑖,𝑗+1−𝑢^𝑛_{𝑖,𝑗−1})−𝑘𝑓(𝑖ℎ_𝑥,𝑗ℎ_𝑦𝑢,𝑛𝑘,^{𝑛𝑖+1,𝑗}+2𝑢^𝑛_𝑖,𝑗+𝑢^𝑛_{𝑖−1,𝑗}+𝑢^𝑛_𝑖,𝑗+1+𝑢^𝑛_{𝑖,𝑗−1}6). (3.3)

Experiment 1. Suppose Ω=[0,2𝜋]×[0,2𝜋], ℎ_𝑥=ℎ_𝑦=2𝜋/40, 𝑘=0.001, 𝑢₀(𝑥,𝑦)=sin𝑥sin𝑦 which itself does not satisfy (1.1); we get the graphs (see Figures 1–3) where 𝑢(𝑥,𝑦,𝑡) changes according to the changes of 𝑡 when different functions are given to 𝑓(⋅,𝑢).

Figure 1 

𝑓(⋅,𝑢)=𝑢.

Figure 2 

𝑓(⋅,𝑢)=sin𝑢.

(a)

(b)

Figure 3 

The numerical results in (a) at 𝑡=0 and in (b) at 𝑡=0.046 when 𝑓(⋅,𝑢)=𝑢2.

Figure 1 shows that when 𝑓(⋅,𝑢)=𝑢, at 𝑡=0.04, the numerical solutions become oscillatory; at 𝑡=0.042, the bifurcation of solutions occurs; when 𝑡>0.042, the solutions will blow-up. Similarly Figure 2 shows that when 𝑓(⋅,𝑢)=sin𝑢, at 𝑡=0.6, the bifurcation of solutions occurs; when 𝑡>0.6, the solutions will blow-up. Figure 3 is the spatiotemporal graphs of solutions when 𝑓(⋅,𝑢)=𝑢² at 𝑡=0 (initial value) and 𝑡=0.0046. When 𝑡>0.0046, the solutions will blow-up.

Experiment 2. The initial value is unknown in the general situation; so we use random numbers ([−0.01,0.01]) as the initial value and draw graphs (see Figures 4 and 5) where 𝑢(𝑥,𝑦,𝑡) changes as 𝑡 changes when different functions are given to 𝑓(⋅,𝑢).

Figure 4 

𝑓(⋅,𝑢)=𝑢.

Figure 5 

𝑓(⋅,𝑢)=1−sin𝑢.

Figures 4 and 5 show that even though the initial value is sufficiently small, the blow-up will appear in finite time for the different functions.

The numerical result shows that there is a locality solution of the equation. When 𝑡 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.

Acknowledgments

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.