Local Existence and Global Existence of differential equations

[machi]

Hi everyone! 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...
thanks before...

 

[Vargo]

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.

 

[machi]

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 :)

 

[Vargo]

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?

 

[machi]

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.