Temperature dependent near field infrared microscopy of La$_{\mathrm{0.67}}$Sr$_{\mathrm{0.33}}$MnO$_{\mathrm{3}}$ thin films

La$_{\mathrm{0.67}}$Sr$_{\mathrm{0.33}}$MnO$_{\mathrm{3}}$ thin films are studied with apertureless, scattering-type near field microscopy at mid-infrared wavelength and varied temperatures. Spatial resolution of about 20 nm is achieved with our technique. The temperature-dependent resistivity shows a continuous second order phase transition between insulating and metallic phases. At most temperatures, near-field infrared microscopy reveals local persistent phase separation that is independent of temperature. It is possible that the local persistent phase separation is induced by strain inhomogeneity in the thin films. Remarkably, we also observe global time-dependent changes in the infrared near-field signal upon repeated scanning of the same microscopic area at a fixed temperature. This observation is consistent with time-dependent, fluctuating conductivity in the vicinity of a second order phase transition.