Consider the intervals $J=[0, \infty)$ and $D=(0,\infty)$ and the initial value problem $\begin{cases} u'(t)=f(t,u(t)) \,\,\,\,\,,\,\, t \in J \\u(0)=2\end{cases}$
where $f:J \times D \to \mathbb R$ , $f(t,v):=-\frac{t}{v} e^{t^2}.$ How can I determine a maximum continued solution of the IVP and check if it is unique?
Some hints are much appreciated.
The standard existence-uniqueness theorem applies to initial value problems where $f$ is such that $\partial f/\partial v$ is continuous. The given $f$ has continuous partial derivatives as long as $v$ does not turn into zero.
To find out how long we have until the solution hits zero, solve the equation. It's separable. To keep things rigorous, let's say that $w=u^2$, and then $$w' = 2uu' = -2te^{t^2} \tag1$$ which leads to $w(t) = C - e^{t^2}$. Here $C=5$ from the initial condition.
Now you know that $u(t) = \sqrt{5-\exp(t^2)}$ for as long as the expression under the root is positive. This is the maximal interval of existence, because if a solution $u$ existed beyond it, the function $u^2$ would be a nonnegative solution of (1), but we know exactly what the solutions of (1) are.