Local Existence and Global Existence of differential equations

In summary, the difference between local and global existence is that local existence only applies to a certain range of t, while global existence applies to all t.
  • #1
machi
3
0
Hi everyone! :smile: I'm newbie in this forum, please help me for my question.

In differential equation we know that the differential equation has a solution and uniqueness. which is usually called the existence and uniqueness theorem. my question, what is the difference of local existence and global existence existence from the point of view of functional analysis?

Please help me with your explanation... :cry:
thanks before... :smile:
 
Physics news on Phys.org
  • #2
Your question is hard to answer without more information. What kind of differential equations are we talking about? ODE, parabolic/hyprebolic PDE?

For problems with a distinguished time variable, the solution is a mapping from R into a space of functions (for PDE). You can think of it as a one parameter family of functions.
Local existence means that that this mapping is defined near 0. Global existence means it extends for all time.
 
  • #3
I forget about it, I mean Ordinary differential Equation in C([a,b],Rn). C([a,b],Rn) is notation for mapping of continuous functions in [a,b] into Rn, I can write f element of C([a,b],Rn) then f=(f1,f2,f3,...,fn).

what do you mean about "the solution is mapping from R into a space of functions" is which have notation C([a,b],Rn)?

for your last sentences, I can understand. thanks :)
 
  • #4
If you have a partial differential equation like the heat equation then your unknown function is something like u(t,x), where t is the time variable and x is the spatial variable. For a fixed t, u(t,x) is the heat distribution over the domain. So a solution of the PDE is viewed as a mapping from the time domain into the domain of heat distributions (which is a space of functions). The reason I thought you might be talking about PDE is because you referred to functional analysis.

I'm still not clear what type of equation you are talking about, but it sounds like an ODE:
y'=F(t,y),
where y is a vector valued function y(t)=(y1(t),... , yn(t)).
In that case, local/global existence refers to whether the solutions are defined for all values of t. Does this answer your original question?
 
  • #5
I think about space of functions in my mind is similar to what you mean. but we have difference on notations. Yes, you are right like your example.

yes, yes, you are right, that was I mean. thak you very much for your help. I would learn more.
 

1. What is the difference between local and global existence of differential equations?

The local existence of a differential equation refers to the existence of a solution within a limited interval of the independent variable. This means that the solution only exists for a specific range of values of the independent variable. On the other hand, the global existence of a differential equation means that the solution exists for all possible values of the independent variable, without any restrictions.

2. How is the local existence of a differential equation determined?

The local existence of a differential equation can be determined through the existence and uniqueness theorem. This theorem states that if the differential equation is well-posed, meaning that it has a unique solution for a given set of initial conditions, then the solution exists locally.

3. What factors can affect the global existence of a differential equation?

The global existence of a differential equation can be affected by the behavior of the solution at the boundaries of the interval, as well as the stability of the solution. If the solution is not stable, it may lead to a blow-up or divergence, causing the global existence to fail.

4. How can the global existence of a differential equation be proven?

To prove the global existence of a differential equation, one must show that the solution exists for all possible values of the independent variable, without any blow-up or divergence. This can be done through mathematical analysis and techniques such as Lyapunov functions and energy methods.

5. Why is the concept of local and global existence important in the study of differential equations?

The concept of local and global existence is important because it helps in determining the validity and stability of solutions to differential equations. It also allows for the prediction of the behavior of a system over time, which is crucial in many scientific and engineering applications.

Similar threads

  • Differential Equations
Replies
5
Views
924
  • Differential Equations
Replies
1
Views
1K
Replies
3
Views
935
Replies
4
Views
706
  • Differential Equations
Replies
5
Views
608
  • Differential Equations
Replies
4
Views
1K
Replies
1
Views
903
  • Differential Equations
Replies
1
Views
707
  • Differential Equations
Replies
12
Views
1K
  • Differential Equations
Replies
1
Views
2K
Back
Top