In this paper, we consider the Cauchy problem to the planar non-resistive magnetohydrodynamic equations without heat conductivity, and establish the global well-posedness of strong solutions with large initial data. The key ingredient of the proof is to establish the a priori estimates on the effective viscous flux and a newly introduced ``transverse effective viscous flux" vector field inducted by the transverse magnetic field. The initial density is assumed only to be uniformly bounded and of finite mass and, in particular, the vacuum and discontinuities of the density are allowed.

Based on a global estimate of the heat kernel, some important inequalities such as Poincaré inequality and log-Sobolev inequality, furthermore a tight logarithm Sobolev inequality are presented on graphs, just under a positive curvature condition CDE'(n,K) with some K > 0. As consequences, we obtain exponential integrability of integrable Lipschitz functions and moment bounds at the same assumption on graphs.

The propagation and interaction in more than one space dimension of localized pulse solutions (so-called light bullets) to the sine-Gordon [SG] equation is studied both asymptotically and numerically. Similar solutions and their resemblance to solitons in integrable systems were observed numerically before in vector Maxwell systems. The simplicity of SG allows us to perform an asymptotic analysis of counterpropagating pulses, as well as a fully resolved computation over rectangular domains. Numerical experiments are carried out on single pulse propagation and on two pulse collision under different orientations. The particle nature, as known for solitons, persists in these two space dimensional solutions as long as the amplitudes of initial data range in a finite interval, similar to the conditions on the vector Maxwell systems.

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We study the front dynamics of the bistable reaction-diffusion equations with periodic diffusion and/or convection coefficients in several space dimensions. When traveling wave solutions exist, the solutions of the initial value problem behave as wave fronts propagating with the effective speeds of traveling waves under various initial conditions. Yet due to the bistable nature of the nonlinearity, traveling waves may not always exist when the medium variations from the mean states are large enough. Their existence is closely related to the detailed forms of diffusion and convection coefficients, more so in multidimension than in one. We present a simple sufficient condition for the nonexistence of traveling waves (quenching) using perturbation method. Our two dimensional finite difference numerical computations show a variety of front behaviors, such as: the propagation, quenching and retreat of fronts. We found