To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the L 2 convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the L 2 norm. Furthermore, the constructed approximate Green function in this paper can be

Let G=(V,E) be a locally finite connected weighted graph, and Δ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation u_t = Δu + f(u) on G. The blow-up phenomenons for u_t = Δu + f(u) are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.

This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier--Stokes equations. Assume that the corresponding Riemann problem to the compressible Euler equations can be solved by rarefaction waves (<i>V^R</i>, <i>U^R</i>, <i>S^R</i>)(<i>t</i>,<i>x</i>). If the initial data (<i>v</i>₀ , <i>u</i>₀ ,<i>s</i>₀ )(<i>x</i>) to the nonisentropic compressible Navier--Stokes equations is a small perturbation of an approximate rarefaction wave constructed as in [S. Kawashima, A. Matsumura, and K. Nishihara, <i>Proc. Japan Acad. Ser. A</i>, 62 (1986), pp. 249--252], then we show that, for the general gas, the Cauchy problem admits a unique global smooth solution (<i>v</i>, <i>u</i>, <i>s</i>)(<i>t</i>,<i>x</i>) which tends to (<i>V^R</i>, <i>U^R</i>, <i>S^R</i>)(<i>t</i>,<i>x</i>) as <i>t</i> tends to infinity. A global stability result can also be established for the nonisentropic ideal polytropic gas, provided that the adiabatic exponent is close to 1. Furthermore, we show that for the