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We consider the large time behavior of solutions to the following nonlinear wave equation:is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if c(·) has a zero point, then c(u(t, x)) can be going to zero in finite time. When c(u(t, x)) is going to 0 in finite time, the equation degenerates. We give a sufficient condition that the equation with 0 ≤ λ < 2 degenerates in finite time. *

Abstract. The initial-value problem for the drift-diffusion equation arising from the model of semiconductor device simulations is studied. The dissipation on this equation is given by the fractional Laplacian (−∆) θ/2 . Large-time behavior of solutions to the drift-diffusion equation with 0 < θ ≤ 1 is discussed. When θ > 1, large-time behavior of solutions is known. However, when 0 < θ ≤ 1, the perturbation methods used in the preceding works would not work. Large-time behavior of solutions to the drift-diffusion equation with 0 < θ ≤ 1 is discussed. Particularly, the asymptotic expansion of solutions with high-order is derived.

Abstract. In this paper, we study blow ups of solutions to the second sound equation, 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) ≥ δ > 0 for some δ. We give sufficient conditions that two types of blow up occur: one of the two types is that L ∞ -norm of ∂ t u or ∂ x u goes up to the infinity; the other type is that u vanishes, that is, the equation degenerates.

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