On a linear model of mechanical phase transitions

Abstract

Existence and infinite speed of propagation are shown for the equation

$$h_n \left( {x,t} \right) = \frac{1}{{\alpha \pi }}\int_{ - \infty }^\infty {\frac{{Ah_{xxx} \left( {\xi ,t} \right) - Bh_{xt} \left( {\xi ,t} \right)}}{{\left( {x - \xi } \right)}}d\xi } , - \infty< x< \infty ,0< t< T,$$

which is the linearization of a model equation for the mechanical phase transition phenomenon of melting-freezing waves. In addition, from the formal solution of the equation one can compute a decay rate.