03 - Maximal Existence Theorem

Maximal Existence Theorem

Let $f:{\mathbb{R}}^2\to \mathbb{R}$ be continuous and locally Lipschitz with respect to $y$
Let $a,b\in \mathbb{R}$ and let $(T_{-},T_{+})$ be the maximal existence interval of solution $y(x)$ for IVP

$$y^{\prime }(x)=f(x,y(x))\text{ and }y(a)=b$$

If $T_{+}<\infty$ then the solutions blows up as we approach $T_{+}$ ($|y(x)|\to \infty$ as $t\to T_{+}$)
If $T_{-}>-\infty$ then $|y(x)|\to \infty$ as $t\to T_{-}$

By thinking of $x$ is a time parameter then we have the following cases

1. Global Existence - Solution of IVP exist for all $x\in \mathbb{R}$
2. Solution exists for all times in future but blows up at some finite time $T_{-}<a$ in the past
3. Solution exists for all times in past but blows up at some finite time $T_{+}>a$ in the future
4. Solution blows up at some finite time in the past and future

Note that can have solutions that exist for all times but tend to infinity as $x\to \pm \infty$

Proof

We only need to prove the case where $T_{+}<\infty$ as we can reverse time direction by

$$\tilde{y}(x)=y(-x)$$

to handle $T_{-}>-\infty$

Suppose that $T_{+}<\infty$ then we need to show that

$$|y(x)|\to \infty \text{ as }x\to T_{+}$$

That is, for every $K>0$ there exists $\delta >0$ such that

$$|y(x)|>K\text{ for all }x\in (T_{+}-\delta ,T_{+})$$

Suppose by contradiction that this isn’t true so there is $K>0$ so that for every $\delta >0$

$$\exists x_{\delta }\in (T_{+}-\delta ,T_{+})\text{ with }|y(\delta )|\le K$$

For some compact rectangle $R_0$ containing set $[-K,K]\times \{T_{+}\}$ then
As $f$ is continuous and $R_0$ is compact then $f$ is bounded on $R_0$ so suppose

$$|f|\le M_0\text{ on }{R}_0$$

Note that the bound still applies for any rectangle $R\subseteq R_0$

So by setting $h:=\min \left\{\frac{1}{M_0},\frac{1}{2}\right\}$ and $k:=1$ so$M_{0}h\le k$ still holds
Hence $P(i)$ holds on

$$R=[\tilde{a}-h,\tilde{a}+h]\times [\tilde{b}-1,\tilde{b}+1]\text{ for all }\tilde{a}\in \left[T_{+}-\frac{1}{2},T_{+}+\frac{1}{2}\right],\tilde{b}\in [-K,K]$$

Since $f$ is locally Lipschitz wrt $y$ on all of ${\mathbb{R}}^2$ then Lipschitz Condition holds on all $R$
Hence we can apply Picard’s Theorem with same $h$ for all initial conditions $y(\tilde{a})=\tilde{b}$

Choosing $\delta <h$ and letting $x_{\delta }\in (T_{+}-\delta ,T_{+})$ so that $|y(x_{\delta })|\le K$
So use $\tilde{a}=x_{\delta }$ and $\tilde{b}=y(x_{\delta })$ as initial data to get solution $\tilde{y}$ of the ODE which satisfies

$$\tilde{y}(x_{\delta })=y(x_{\delta })$$

which is defined on $[x_{\delta }-h,x_{\delta }+h]$

As $x_{\delta }+h>T_{+}-\delta +h>T_{+}$ which allows us to extend the original solution $y$
beyond the maximal existence time $T_{+}$ which is a contradiction