  We solve the Maxwell and heat equations self-consistently for metal
nanoparticles under intense continuous wave (CW) illumination. Unlike previous
studies, we rely on {\em experimentally}-measured data for the metal
permittivity for increasing temperature and for the visible spectral range. We
show that the thermal nonlinearity of the metal can lead to substantial
deviations from the predictions of the linear model for the temperature and
field distribution, and thus, can explain qualitatively the strong nonlinear
scattering from such configurations observed experimentally. We also show that
the incompleteness of existing data of the temperature dependence of the
thermal properties of the system prevents reaching a quantitative agreement
between the measured and calculated scattering data. This modelling approach is
essential for the identification of the underlying physical mechanism
responsible for the thermo-optical nonlinearity of the metal and should be
adopted in all applications of high temperature nonlinear plasmonics,
especially for refractory metals, both for CW and pulsed illumination.
