We consider a model semilinear reaction-diffusion system with cubic nonlinear reaction terms and small spatially decaying initial data on R 1. The model system is motivated by the thermal-diffusive system in combustion, and it reduces to a scalar reaction-diffusion equation with Zeldovich nonlinearity when the Lewis number is one and proper initial data are prescribed. For scalar equations of similar type it is well known that while a nonlinearity of degree greater than three (supercritical case) has no effect for large times a cubic nonlinearity qualitatively changes the long time behaviour. The latter case has been treated in the literature by a rescaling method under the additional assumption of smallness of the nonlinearity. Although for our system the cubic nonlinearity is also critical we establish large time behaviour when the nonlinearity is not necessarily small which essentially differs from the supercritical case. This

The formation and propagation of thermal fronts in a cylindrical medium that is undergoing microwave heating is studied in detail. The model consists of Maxwell's wave equation coupled to a temperature diffusion equation containing a bistable nonlinear term.

Despite its usefulness, the Kalman-Bucy filter is not perfect. One of its weaknesses is that it needs a Gaussian assumption on the initial data. Recently Yau and Yau introduced a new direct method to solve the estimation problem for linear filtering with non-Gaussian initial data. They factored the problem into two parts: (1) the on-line solution of a finite system of ordinary differential equations (ODEs), and (2) the off-line calculation of the Kolmogorov equation. Here we derive an explicit closed-form solution of the Kolmogorov equation. We also give some properties and conduct a numerical study of the solution.

We consider a family \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} of \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} -periodic Schrdinger operators with \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} -interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} surfaces. We show that in the limit when \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} and the interactions strengths are appropriately scaled, \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} has at most \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} gaps within finite intervals, and moreover, the limiting behavior of the first \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.

In this paper, we consider the local existence of solutions to Euler equations with linear damping under the assumption of physical vacuum boundary condition. By using the transformation introduced in Lin and Yang (Methods Appl. Anal. 7 (3) (2000) 495) to capture the singularity of the boundary, we prove a local existence theorem on a perturbation of a planar wave solution by using LittlewoodPaley theory and justifies the transformation introduced in Liu and Yang (2000) in a rigorous setting.