Consider the ODE given by, $x'=t^2e^{nx^2}$, and initial condition $x(t_0)=0$, $t_0 \in \mathbb{R}, n\in \mathbb{Z}$. They ask me about the existence and uniqueness of solutions for this Cauchy problem. I also have to determine the interval of definition of the maximal solution for $n\leq0$.
My approach,
If I show that $f(t,x)=t^2e^{nx^2}$ is continous and Lipschitz with respect to the second variable, by Picard's theorem, I can assure that there exists a unique solution to the given Cauchy problem. So let's try to do so,
Let's first show that $f(t,x)$ is continous.
$$f(t_0,0)=t_0^2$$
If we define, $t:=rcos\theta + t_0$, and $x:=rsin\theta$, then $$\lim_{r\rightarrow 0}(rcos\theta + t_0)^2e^{nr^{2}sin^{2}\theta} = t_0^2$$
So $f(t,x)$ is continous. Let's try to prove that it is Lipschitz.
For $f$ to be Lipschitz it would have to verify that, $$||f(t,x)-f(t,y)||\leq K|x-y|$$ for some $K>0$.
$$||f(t,x)-f(t,y)||=||t^2e^{nx^2}-t^2e^{ny^2}||=t^2|e^{nx^2}-e^{ny^2}|\leq 2n\zeta t^2e^{n\zeta ^2}|x-y|$$ And I don't know if $0<K<2n\zeta t^2e^{n\zeta ^2}$, given the domain, $t_0\in \mathbb{R}, n\in \mathbb{Z}, \zeta \in (x,y)$.
My questions, which I hope someone can help me solve are, What can I say about the existence and uniqueness of solutions for this Cauchyy Problem? Is there any other criterion that assures existence but not uniqueness? What is the interval of definition of the maximal solution if $n\leq 0$?