  We propose a theory of low temperature thermal transport in nano-wires in the
regime where a competition between phonon and flexural modes governs the
relaxation processes. Starting with the standard kinetic equations for two
different types of quasiparticles we derive a general expression for the
coefficient of thermal conductivity. The underlying physics of thermal
conductance is completely determined by the corresponding relaxation times,
which can be calculated directly for any dispersion of quasiparticles depending
on the size of a system. We show that if the considered relaxation mechanism is
dominant, then at small wire diameters the temperature dependence of thermal
conductivity experiences a crossover from $T^{1/2}$ to $T^3$-dependence.
Quantitative analysis shows reasonable agreement with resent experimental
results.
