# 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