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This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one-dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis.

In this paper, we establish the blow-up theorems of Fujita type for a class of homogeneous Neumann exterior problems of quasilinear convection–diffusion equations. The critical Fujita exponents are determined and it is shown that the exponents belong to the blow-up case under any nontrivial initial data. Copyright © 2010 John Wiley & Sons, Ltd.

Mush-chimney convection occurs in a system consisting of a layer of mixed solid and liquid bounded by liquid above and solid below. The solid phase of the mush is composed of the denser constituent of the alloy, making the residual liquid compositionally buoyant as solidification proceeds. The liquid is in steady, nonlinear convection, consisting of a downward flow in the mush and an upward flow in a chimney, free of solid, which formed spontaneously in the mush as a result of the interaction between flow and structure. A well-posed continuum mathematical model of this system is developed. A novel feature of this analysis is the development of an equation that determines the chimney radius. Material diffusion is an essential ingredient in this equation, but is negligible elsewhere. In the parameter range considered, the problem is well posed without the necessity of specifying the condition of marginal phase equilibrium. A remarkably simple analytic solution is obtained for the convective flow in the chimney.