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Maximal solution to Cauchy problems

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The maximal solution to Cauchy problems is a solution that gives full information on the physical model. In fact, we obtain this solution by extending local solutions. In this post, we give some theorem that shows the existence of the uniqueness of the maximal solution.

What is the maximal solution to Cauchy problems?

Let $f: I\times \Omega\to \mathbb{R}^b$ be a continuous function, where $I$ is an interval of $\mathbb{R}$ and $Omega$ an open set of $\mathbb{R}^d$. Moreover, let $(t_0,x_0)\in I\times \Omega$ and consider the Cauchy problem\begin{align*} (CP)\quad\begin{cases}\dot{u}(t)=f(t,u(t)),& t\in I,\cr u(t_0)=x_0.\end{cases}\end{align*}

A maximal solution of the Cauchy problem (CP) that can not admits an extension to another solution.

Peano’s theorem gives the existence of the maximal solution under the continuity assumption of the vector field $f$ on $I\times \Omega$. On the other hand, if in addition, we assume that $f$ is locally Lipschitz with respect to its second variable, then the Cauchy problem admits a unique maximal solution. This result is called the Cauchy-Lipschitz theorem.

We recall that a function $f$ is locally Lipschitz with respect to its second variable, if for any $(t_0,x_0)\in I\times \Omega$ there exists a neighborhood $V_{t_0,x_0}$ of $(t_0,x_0) $ and a constant $C>0$ such that for any $(t,x)$ and $(s,y)$ in $V_{t_0,x_0}$ we have \begin{align*}\|f(t,x)-f(t,y)\|\le C \|x-y\|.\end{align*}

We mention that the maximal solution is necessarily defined on an open interval of the form $(\alpha,\beta)$.

The global solutions to Cauchy problems

A solution $u: J\to \Omega$ of the Cauchy problem (CP) is called global if $J=I$. It is then important to look for conditions that guarantee the existence of global solutions. In fact, there exists a nice theorem, called the explosion theorem, that gives these conditions.

Assume that $I=(a,b)$ with $a$ can possibly be $-\infty$, and $b$ can also be $+\infty$. In addition, we suppose that $\Omega=\mathbb{R}^d$. Let $u:(\alpha,\beta)\to\mathbb{R}^d$ be a maximal solution. Then we have the following cases:

  • $\beta=b$, if not then $|u(t)|\to+\infty$ as $t\to \beta$. This means that if the solution is not left global then it is unbounded in a neighborhood of $\beta$.
  • $\alpha=a$, if not $|u(t)|\to+\infty$ as $t\to \alpha$. This means that if the solution is not right global, then it is unbounded in a neighborhood of $\alpha$.
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