Prolongation of solutions of differential equations

The property of solutions of ordinary differential equations to be extendible to a larger interval of the independent variable. Let

$$ \tag{1 } x = \phi ( t),\ \ t \in I, $$

be a solution of the system

$$ \tag{2 } \dot{x} = f ( t, x),\ \ x \in \mathbf R ^ {n} . $$

A solution $ x = \psi ( t) $, $ t \in J $, is called a prolongation of the solution (1) if $ J \supset I $ and $ \psi ( t) \equiv \phi ( t) $ for $ t \in I $.

Suppose that the function

$$ f ( x, t) = ( f _ {1} ( t, x _ {1} \dots x _ {n} ) \dots f _ {n} ( t, x _ {1} \dots x _ {n} )) $$

is defined in a domain $ G \subset \mathbf R _ {t,x} ^ {n + 1 } $ and suppose $ t _ {0} \in I $. The solution (1) is called indefinitely extendible (indefinitely extendible forwards (to the right), indefinitely extendible backward (to the left)) if a prolongation of it exists defined on the axis $ - \infty < t < \infty $( respectively, on the semi-axis $ t _ {0} \leq t < \infty $, on the semi-axis $ - \infty < t \leq t _ {0} $). The solution (1) is called extendible forwards (to the right) up to the boundary $ \Gamma $ of $ G $ if a prolongation $ x = \psi ( t) $, $ t _ {0} \leq t \leq t _ {+} < \infty $, of it exists with the following property: For any compact set $ F \subset G $ there is a value $ t = t _ {F} $, $ t _ {0} < t _ {F} < t _ {+} $, such that the point $ ( t _ {F} , \psi ( t _ {F} )) $ does not belong to $ F $. Extendibility backward (to the left) up to the boundary $ \Gamma $ is defined analogously. A solution that cannot be extended is called non-extendible.

If the function $ f ( t, x) $ is continuous in $ G $, then every solution (1) of (2) can be either extended forwards (backward) or indefinitely or up to the boundary $ \Gamma $. In other words, every solution of (2) can be extended to a non-extendible solution. If the partial derivatives

$$ \tag{3 } \frac{\partial f _ {i} }{\partial x _ {j} } ,\ \ i, j = 1 \dots n , $$

are continuous in $ G $, then such a prolongation is unique.

An interval $ J $ is called a maximal interval of existence of a solution of (2) if the solution cannot be extended to a larger interval. For any solution of a linear system

$$ \dot{x} _ {i} = \ \sum _ {j = 1 } ^ { n } a _ {ij} ( t) x _ {j} + f _ {i} ( t),\ \ 1 \leq i \leq n, $$

with coefficients $ a _ {ij} ( t) $ and right-hand sides $ f _ {i} ( t) $, $ 1 \leq i, j \leq n $, that are continuous on an interval $ J $, the maximal interval of existence of a solution coincides with $ J $. For solutions of a non-linear system the maximal intervals of existence may differ for different solutions, and determining them is a difficult task. E.g. for the solution to the Cauchy problem

$$ \dot{x} = x ^ {2} ,\ \ x ( t _ {0} ) = x _ {0} , $$

one has

$$ J = ( t _ {0} + x _ {0} ^ {-} 1 , \infty ) $$

if $ x _ {0} < 0 $,

$$ J = (- \infty , t _ {0} + x _ {0} ^ {-} 1 ) $$

if $ x _ {0} > 0 $, and

$$ J = (- \infty , \infty ) $$

if $ x _ {0} = 0 $.

A sufficient condition under which one can indicate the maximal interval of existence of a solution is, e.g., Wintner's theorem: Suppose that the function $ f ( t, x) $ is continuous for $ t \in J = [ t _ {0} , t _ {0} + a] $, $ x \in \mathbf R ^ {n} $, and that it satisfies in this domain the estimate

$$ | f ( t, x) | \leq L ( | x | ), $$

where $ L ( r) $ is a function continuous for $ r \geq 0 $, $ L ( r) > 0 $ and for some $ \delta $, $ 0 \leq \delta < \infty $,

$$ \int\limits _ \delta ^ \infty \frac{dr }{L ( r) } = + \infty . $$

Then every solution of (2) exists on the whole of $ J $.

This theorem also holds for $ J = [ t _ {0} , \infty ) $. Sufficient conditions for indefinite extendibility of a solution are of great interest. E.g., if $ f ( t, x) $ and its partial derivatives (3) are continuous for $ t _ {0} \leq t < \infty $, $ x \in \mathbf R ^ {n} $, and if for these values of $ t, x $ the estimates

$$ \left | \frac{\partial f _ {i} }{\partial x _ {j} } \right | \leq c ( t) < \infty ,\ \ i, j = 1 \dots n, $$

hold, then the solution of (2) with $ x ( t _ {0} ) = x _ {0} $ exists for $ t _ {0} \leq t < \infty $, for any $ x _ {0} \in \mathbf R ^ {n} $.

Consider the Cauchy problem

$$ \tag{4 } \dot{x} = f ( x),\ \ x ( t _ {0} ) = x _ {0} , $$

for an autonomous system, where $ f ( x) $ is continuously differentiable in a domain $ G \subset \mathbf R _ {x} ^ {n} $. If, as $ t $ grows, the phase trajectory of the solution $ x = \phi ( t) $ of (4) remains in a compact subset $ F \subset G $, then this solution can be extended to the semi-axis $ t _ {0} \leq t < \infty $.

References

[1]	L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)
[2]	V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian)
[3]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[4]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[5]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[6]	L. Cesari, "Asymptotic behavior and stability problems in ordinary differential equations" , Springer (1959)
[7]	A. Wintner, "The non-local existence problem of ordinary differential equations" Amer. J. Math. , 67 (1945) pp. 277–284

Comments

Instead of prolongation of solutions, continuation of solutions is nowadays mostly used.