What is a sublinear function? Is $y'(x) = x^5(e^{4-y^2}-1)$ sublinear?

Question

May I have any simplest definition of sublinear function? I tried reading through Wikipedia but couldn't understand it well. Moreover, how can I check whether any function follows sublinearity or not? Also, does sublinearity implies global Lipschitzness?

Answer 1

Yes. It is even bounded in $y$. So you get $$ |y'|\le |x|^5(e^4-1) $$ which implies $$ |y(x)-y(0)|\le \frac{|x|^6}6(e^4-1). $$ This is the kind of bound one can use to show that the maximal domain is $\Bbb R$.

In general an ODE is sub-linear if $|f(x,y)|\le K(x)+L(x)|y|$. In a slightly restricted but more simple version one can also find $|f(x,y)|\le L|y|$ for $|y|>R$ for some radius $R$ and constant $L$.