I'm currently stuck on the following problem:
Let $n\in\mathbb N$ be a fixed natural number $t\mapsto \left(x(t),y(t)\right)$ be one of the solutions to the differential system \cases{x'=e^{-y^2}\sin(x^n+y^n),\\ y'=x^n\sin(x^n+y^n).} Then prove that all such maps are defined on $[0,+\infty).$
I don't know how to conclude neatly the problem I admit, in particulare my plan is focused on the condition of strict decrease of x'. Then one would like to show that if the maximal interval of definiton were of the form $[0,\alpha),\alpha<+\infty$, then we would find a compact subset which contains the solution. But I cannot delineate it neatly, as i said, so any help is welcomed. Thank you and regards.