Power type

A nonlinear function $F:\mathbb {R} ^{n}\to \mathbb {R} ^{m}$ is of power type with exponent $p>1$ if one has the bounds

$|F(u)|\leq C|u|^{p}$

(so in particular $F(0)=\nabla F(0)=0$) and

$|F(u)-F(v)|\leq C|u-v|(|u|^{p-1}+|v|^{p-1})$

for all $u,v\in \mathbb {R} ^{n}$ and some constant $C>0$. Note that the first bound is a special case of the second once one assumes that $F(0)=0$. If F is continuously differentiable, then the second bound is equivalent to the bound

$|\nabla F(u)|\leq C|u|^{p-1}$

(possibly for a slightly different value of C), thanks to the fundamental theorem of calculus identity

$F(u)-F(v)=\int _{0}^{1}(u-v)\cdot \nabla F((1-t)u+tv)\ dt.$

The model example of a power type nonlinearity is the pure power nonlinearity $F(u)=\pm |u|^{p-1}u$ (for either real or complex u). If p is an integer, any function F which is a homogeneous polynomial of degree p in u and (in the complex case) ${\overline {u}}$ also qualifies as a power type nonlinearity. As a rule of thumb, local well-posedness results which hold for pure power nonlinearities, also hold for power type nonlinearities of the same exponent. However, in the high regularity theory, it is often necessary to impose additional hypotheses on F, for instance that $\nabla F$ is a power type nonlinearity of order p-1.