We consider the Cauchy problem associated with $F_0$ defined by: $$\forall y \in ]0, +\infty[, \quad F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$$ with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$. We denote $\phi_0$ the solution of this problem on $[0, +\infty[$. (a) Show that there exists $\varepsilon > 0$ such that for all $t \in ]0, \varepsilon]$ we have $y_{\text{init}} < \phi_0(t) < \theta$. (b) By considering the function $z_0(t) = \ln\left(\phi_0(t)/\theta\right)$ find the expression of $\phi_0$. (c) Deduce that $\phi_0$ satisfies $y_{\text{init}} < \phi_0(t) < \theta$ for all $t \in ]0, +\infty[$ and that moreover $\phi_0$ is strictly increasing.
For $0 < \mu \leqslant 1$, we consider $F_\mu$ defined by: $$\forall y \in ]0, +\infty[, \quad F_\mu(y) = \frac{a}{\mu} y\left(1 - \left(\frac{y}{\theta}\right)^\mu\right)$$ with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$. By considering the function $z_\mu(t) = \phi_\mu(t)^{-\mu}$ find the expression of the solution $\phi_\mu$ on $[0, +\infty[$ associated with $F_\mu$.
For $0 < \mu \leqslant 1$, we consider $F_\mu$ defined by: $$\forall y \in ]0, +\infty[, \quad F_\mu(y) = \frac{a}{\mu} y\left(1 - \left(\frac{y}{\theta}\right)^\mu\right)$$ and $F_0$ defined by $F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$, with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$. (a) Show that $F_\mu$ converges pointwise to $F_0$ as $\mu$ tends to 0. (b) Show that $\phi_\mu$ converges pointwise to $\phi_0$ as $\mu$ tends to 0.
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. For all $r \geqslant 0$, we denote $B_r$ the closed ball with center $y_{\text{init}}$ and radius $r$. Show that we can choose $r > 0$ and $T > 0$ such that $B_r \subset \Omega$ and such that for all $N \in \mathbb{N}^*$, we can define by recursion, by setting $\Delta t = \frac{T}{N}$, a sequence $(y_n)_{0 \leqslant n \leqslant N}$ taking values in $B_r$ such that: $$y_0 = y_{\text{init}}, \quad y_{n+1} = y_n + \Delta t F(y_n), \forall n \in \{0, \cdots, N-1\}$$
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With $r > 0$, $T > 0$, $\Delta t = \frac{T}{N}$ and the sequence $(y_n)_{0 \leqslant n \leqslant N}$ as defined in question III.1, show that we can construct a unique function $\phi_N$ continuous on $[0, T]$, affine on each interval $[n\Delta t, (n+1)\Delta t]$ for all $n \in \{0, \cdots, N-1\}$ and such that $\phi_N(n\Delta t) = y_n$ for all $n \in \{0, \cdots, N\}$.
QIII.3
First order differential equations (integrating factor)Existence ProofView
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With the functions $\phi_N$ constructed in question III.2, show, using Theorem 1, that there exists a subsequence of $\phi_N$ that converges uniformly on $[0, T]$ to a continuous function $\phi$.
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. Show that we can define a sequence of step functions $\psi_N : [0, T] \rightarrow \mathbb{R}^d$ such that $\psi_N(t) = \phi_N(t)$ for $t \in \{n\Delta t \mid n \in \{0, \cdots, N\}\}$ and such that: $$\forall N \in \mathbb{N}^*, \phi_N(t) = y_{\text{init}} + \int_0^t F(\psi_N(s))\, ds \text{ for all } t \in [0, T]$$ We will specify the expression of the functions $\psi_N$.
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With the step functions $\psi_N$ defined in question III.4, deduce that there exists a subsequence of $\psi_N$ that converges uniformly on $[0, T]$ and specify its limit.
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. Show that $(\phi, T)$ is a solution of the Cauchy problem $$\left\{\begin{array}{l} y'(t) = F(y(t)) \\ y(0) = y_{\text{init}} \end{array}\right.$$ and deduce the following theorem: Theorem 2: If $F$ is a continuous function, then there exists at least one solution of the Cauchy problem (1).
We consider the particular case for $d = 1$ given for all $y \in \mathbb{R}$ by $F(y) = 3|y|^{2/3}$ and $y_{\text{init}} = 0$. Show that this Cauchy problem admits infinitely many global solutions.