BLOW UP OF SOLUTIONS TO THE SECOND SOUND EQUATION IN ONE SPACE DIMENSION

In this paper, we study blow ups of solutions to the second sound equation partial derivative(2)(t)u = u partial derivative(x)(u partial derivative(x)u), which is more natural than the second sound equation in Landau-Lifshitz's text in large time. We assume that the initial data satisfies u (0, x) >= delta > 0 for some delta. We give sufficient conditions that two types of blow up occur: one of the two types is that L-infinity-norm of partial derivative(t)u or partial derivative(x)u goes up to the infinity; the other type is that u vanishes, that is, the equation degenerates.