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Let G = (V, E) be a finite or locally finite connected weighted graph, ∆ be the usual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on GWe conclude that, for a graph satisfying curvature dimension condition CDE (n, 0) and V (x, r) r m , if 0 < mα < 2, then the non-negative solution u is not global, and if mα > 2, then there is a non-negative global solution u provided that the initial value is small enough. In particular, these results are true on lattice Z m .

The reflection dispersion relation and emission spectra of Ag∕SiO2∕Ag trilayer plasmonic thermal emitters with different lattice constant and Ag line width were investigated. The top Ag film is perforated with parallel line-shape hole array. The induced Ag∕SiO2 surface plasmons at both top and bottom Ag∕SiO2 interface are found coupled together. The coupling effect results in the localized surface plasmon polaritons confined at the top Ag∕SiO2 interface which exhibit the Fabry–Pérot resonance. The thermal emission peak position coincides with the reflection minimum in the dispersion relation and shifts to long wavelength as the Ag line width increases, which proves that the emission is due to the excitation of localized surface plasmon polaritons. Moreover, the emitted light is polarized perpendicular to the parallel metal lines.