Lipschitz condition and blow up time

In summary, the Lipschitz condition is a sufficient condition for existence, uniqueness, and continuous dependence of solutions to first order differential equations in one dimension. However, this condition does not guarantee that the solution will exist for all time. Counterexamples exist, such as x' = x^2 and x' = -x^3, where the solutions blow up at finite times. Further investigation into these counterexamples can provide a better understanding of the conditions under which the solution will blow up in finite time.
  • #1
csco
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Hello everyone, I have asked a similar question in the DE forum but couldn't get an answer so I'm hoping the mods will be tolerant and let me post it here even though it's not strictly analysis.

I'm considering a DE of the form x' = f(t, x) where f is a continuous function defined on an open set D of R2. More importantly f is locally Lipschitz with respect to x which implies there exists a unique maximal solution defined in the maximal interval I(t0, x0) to the initial value problem x' = f(t, x), x(t0) = x0. In short these are first order differential equations in one dimension which verify existence, uniqueness and continuous dependence with respect to initial conditions.

Now for nonlinear equations like x' = x2 the solution cannot be extended beyond a certain time (the maximal interval I(t0, x0) is not all R). The solution to x' = x^2, x(t0) = x0 is x(t) = x0/(1 +x0(t0 - t)) which blows up when t = 1/x0 + t0. In this case the blow up time is a continuous function of the initial conditions. My question is if the Lipschitz condition is enough to ensure this always happens or if it isn't and what would be a counterexample in that case.

I'm hoping someone can at least tell me if this is a too difficult problem and I'm wasting time with it or if I should look for a counterexample or a proof of the above.
 
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  • #2




Thank you for your question. As you have correctly stated, the Lipschitz condition is a sufficient condition for existence, uniqueness, and continuous dependence of solutions to first order differential equations in one dimension. However, this condition does not guarantee that the solution will exist for all time. In fact, there are counterexamples where the solution will blow up in finite time, as you have mentioned in your post.

One such counterexample is the equation x' = x^2, x(0) = 1. The solution to this equation is x(t) = 1/(1-t), which blows up at t = 1. This is a continuous function of the initial condition, as you have pointed out.

Another example is the equation x' = -x^3, x(0) = 1. The solution to this equation is x(t) = 1/sqrt(1+2t), which blows up at t = -1/2. Again, this is a continuous function of the initial condition.

In general, the Lipschitz condition does not guarantee that the solution will exist for all time. The existence of a finite blow up time depends on the specific form of the equation and the initial condition. Therefore, it is not a too difficult problem and it is worth investigating further. I suggest looking for counterexamples and studying their behavior to gain a better understanding of the conditions under which the solution will blow up in finite time.

I hope this helps and good luck with your research.


 

1. What is the Lipschitz condition?

The Lipschitz condition is a mathematical concept that describes the smoothness or regularity of a function. It states that for a function to be Lipschitz continuous, there must exist a constant value such that the absolute value of the difference in the function's output for any two points is less than or equal to the constant multiplied by the distance between those two points.

2. How does the Lipschitz condition relate to blow up time?

The Lipschitz condition is closely related to blow up time in the study of partial differential equations. If a function satisfies the Lipschitz condition, it means that it is locally well-behaved and has a finite derivative at each point. This allows for the analysis of the function's behavior over time, including the prediction of potential blow up or divergence.

3. What is blow up time?

Blow up time is a term used in the study of partial differential equations to describe the moment when a solution to an equation becomes infinite or undefined. It is often used to determine the stability of a system and can provide important insights into the long-term behavior of a solution.

4. How is the Lipschitz condition tested in practice?

In practice, the Lipschitz condition can be tested by calculating the derivative of a function at different points and comparing it to a constant value. If the derivative exceeds the constant value, then the function does not satisfy the Lipschitz condition and may be prone to blow up or divergence.

5. Can the Lipschitz condition be relaxed for certain functions?

Yes, the Lipschitz condition can be relaxed in certain cases. For example, if a function is only required to be Lipschitz continuous in a bounded region, it may still be considered well-behaved and have a finite derivative in that region. This allows for a wider range of functions to be analyzed while still providing useful insights through the Lipschitz condition.

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