Cauchy initial value problem

Let $D$ be a subset of ${ℝ}^n\times ℝ$, $(x_0,t_0)$ a point of $D$, and $f:D\to ℝ$ be a function.

We say that a function $x(t)$ is a solution to the Cauchy (or initial value) problem

$$\{\begin{array}{cc}{x}^{\prime }(t)=f(x(t),t)\hfill & \\ x(t_0)=x_0\hfill & \end{array}$$ (1)

if

1. $x$ is a differentiable function $x:I\to {ℝ}^n$ defined on a interval $I\subset ℝ$;

2. one has $(x(t),t)\in D$ for all $t\in I$ and $t_0\in I$;

3. one has $x(t_0)=x_0$ and $x^{\prime }(t)=f(x(t),t)$ for all $t\in I$.

We say that a solution $x:I\to {ℝ}^n$ is a maximal solution if it cannot be extended to a bigger interval. More precisely given any other solution $y:J\to {ℝ}^n$ defined on an interval $J\supset I$ and such that $y(t)=x(t)$ for all $t\in I$, one has $I=J$ (and hence $x$ and $y$ are the same function).

We say that a solution $x:I\to {ℝ}^n$ is a global solution if $D\subset ={ℝ}^n\times I$.

We say that a solution $x:I\to {ℝ}^n$ is unique if given any other solution $y:I\to {ℝ}^n$ one has $x(t)=y(t)$ for all $t\in I$ (i.e. $x$ is the unique solution defined on the interval $I$).

0.1 Notation

Usually the differential equation in (1) is simply written as $x^{\prime }=f(x,t)$. Also, depending on the topics, the name chosen for the function and for the variable, can change. Other common choices are $y^{\prime }=f(y,t)$ or $y^{\prime }=f(y,x)$. It is also common to write $\dot{x}=f(x,t)$ when the independent variable represents a time value.

0.2 Examples

1. The function $x(t)=\log t$ defined on $I=(0,+\mathrm{\infty })$ is the unique maximal solution to the Cauchy problem:

   $$\{\begin{array}{cc}{x}^{\prime }(t)=1/t\hfill & \\ x(1)=0.\hfill & \end{array}$$

   In this case $f(x,t)=1/t$, $D=\{(x,t):t\ne 0\}$, $t_0=1$, $x_0=0$.

2. The function $x(t)=e^t$ is a global (and hence maximal), unique solution to the Cauchy problem:

   $$\{\begin{array}{cc}{x}^{\prime }(t)=x(t)\hfill & \\ x(0)=1.\hfill & \end{array}$$

3. Consider the Cauchy problem

   $$\{\begin{array}{cc}{x}^{\prime }(t)=\frac{3}{2}\sqrt[3]{x}\hfill & \\ x(0)=0.\hfill & \end{array}$$

   The function $x(t)=0$ defined on $I=ℝ$ is a global solution. However the function $y(t)=\sqrt{t^3}$ defined on $I=[0,+\mathrm{\infty })$ is also a solution and so are the functions

   for every $c\ge 0$. So there are no unique solutions. Moreover $y$ is not a maximal solution.

Title	Cauchy initial value problem
Canonical name	CauchyInitialValueProblem
Date of creation	2013-03-22 14:57:18
Last modified on	2013-03-22 14:57:18
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	14
Author	paolini (1187)
Entry type	Definition
Classification	msc 34A12
Synonym	Cauchy problem
Synonym	initial value problem
Related topic	InitialValueProblem
Related topic	DifferentialEquation
Related topic	CauchyKowalewskiTheorem
Defines	solution to the Cauchy problem
Defines	solution to the initial value problem