Blow up Property of the Solution for Quasi Linear Parabolic Equations and Its Application

This dissertation is to discuss the initial-boundary value problem under the third nonlinear boundary condition for a kind of quasi-linear parabolic equations. To apply the maximum value theory and convex method, it is proved that the blowing up of solution in the definite time. And for application, this paper research into a mathematics model of fluids in porous medium. And have got the blowing up behavior for the problem in limited time.


Introduction
It is significant practically for the research that the solution of mathematics physical partial differential coefficient equation (Developmental Equation) which is related to the time variable t and appear in engineering dynamics such as the limited swing wave spreading, mixed gas burning, fluent mechanics and reaction diffusing, blows up in limited time.Because, whether the research of unitary characters (e g. stability) of fixed solution problem for nonlinear developmental equation or finding the methods of numerical value computing, both base on unitary existence of the solution.If the obtained solution blows up in limited time (the solution is to infinite in limited time), however, such blowing up behavior is not permit by the relevant engineer dynamics model, this shows the engineer dynamics model is questionable and it should be modified.If such blowing up behavior permit the studied engineer dynamics problem, we must compute in a more extensive function since the relevant engineering processes will keep on developing by no means of ending at a certain moment.
This dissertation discuss the original boundary value problem for the quasi-lineal parabolic equation with the third type nonlinear boundary condition: ). Dissertation (Pao C.V., 1980) is to discuss the blowing up character for the solution of problem (1) while f = λ(e au − b) (where λ, a, b are non-negative constant) and g = 0. Suppose b(x) ≡ 0 and f (u) is just the function of u, literatures (Deng Jucheng, 1987;Chipot M., 1989) studies when g = 0 and g = g(u), solution of problem (1) blow up in limited time; on the bases of b(x) 0 and f is a function of x, t, and u , literature (Zheng Zongmu, 1987) discusses blow up problem of boundary condition g ≡ 0 .The problem discussed in literatures (Gomez J.L., 1991;Migoguchi N., 1997;Zhang Hailiang, 2002;Li Junfeng, 2002) is special situation of (1).This dissertation extends the problem studied in literature (Zha Zhongwei, 1992), accordingly original boundary value problem has more comprehensive physical backgrounds.Suppose: , g t ≥ 0, quantitative product g q • q t is non-negative; when p ≥ 0, g ≤ 0; when p < 0, g > 0. to the same h, when p ≥ 0, it has the inequality g p < 1 and g

The non-negative character of solution
In order to prove the solution of problem (1) blowing up in limited time, we should apply some lemmas as follows.
Obviously, we just need to prove solution of problem (3) V(x, t) is non-negative on D × [0, T ).To apply counter evidence, providing V(x, t) can take negative, then it must have a point M 0 (x 0 , t 0 ) result in V(M 0 ) the negative minimum value , according to (3) V(x, 0) = u 0 (x) to know t 0 0.
), we can know the right of (3)' first equation should be positive, which is illogical.
If M 0 ∈ ∂D × (0, T ), according to the maximum value principle we know ∂V ∂γ | M 0 ≤ 0, the left of second equation in (3) is negative here, which is illogical with condition A3).So, it must have u(x, t) ≥ 0 on D × [0, T ) .

With the available condition of A1)-A4), if unitary slick solution of original boundary value problem
Proof We just need to prove existing limited moment T 0 , result in the solution of question (4) W(x, t) blow up at T 0 .That is There into, h > 1 is the constant in condition A2) .Finding differential coefficient with both sides of equation ( 7), we obtain Through lemma 1, lemma 2 to know To take first equation of ( 4) into ( 8), then Applying subsection integral to the first integral of the right side of above equation, we obtain: there into ds is the area element of ∂D .To find differential coefficient with equation ( 11) concerning t, we obtain In addition, if to find differential coefficient with both sides of (8) directly, we obtain To take equation of ( 6) into ( 13) and apply subsection integral, we obtain Using 2 to multiply ( 14) and then subtract (12), with attention to the boundary condition in the ( 6) then obtain According to supposed condition A1)-A4) and lemma1, lemma2 concerning the conclusion of non-negative W, ∂W ∂t , we know then through ( 9), (10) to know On the other side, through in equation ( 15) to know J (t) ≤ 0, According to Lemma 3, exists T 0 (0 < T 0 < − J(0) J (0) ), cause J(T 0 ) = 0, we obtain lim theorem has been proved.

Applied example
In the J. Bear's specialized work (Bcar., 1972), he proposed that the law of fluids in porous medium can be concluded to a quasi linear parabolic equation u t = (u m )(m > 1) We can put the u < 0 situation aside because it does not exist in real life.Unfolding the equation and considering the initial boundary value problem: The discussion below is to seek the conditions to the blowing up appearance of the solution of (17).
By the supposed condition A2) it requests, By the condition A3), it requests, g ≤ 0, g t ≥ 0 and g u + (h − 1)u − 1 = (h − 1)u − 1 ≥ 0 then we get u ≥ 1 h−1 .Considering it with (18) we get Finality with the premise of A4), we get m(e u 0 ) m−1 (e u 0 ) + m(m − 1)(e u 0 ) m−2 (∇ x e u 0 ) 2 = m(e u 0 ) m [m(∇ x u 0 ) 2 + u 0 ] ≥ 0 obviously u 0 ≥ 0 is the only prerequisite.According the discussion above we get the conclusion: If the initial boundary of problem(4.1)given functions satisfies g ≤ 0, g t ≥ 0 and u 0 ≥ 0. u(x, t) is the solution of ( 17) when it satisfies the inequalities of ( 18), (20).Then there must be definite time T 0 (0 < T 0 < +∞), it makes blow up at T 0 .The blowing up appearance help us to think about that the model of fluids low is reasonable and solvable.