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Analytical solutions to heat or diffusion type equations are numerous, but there are rather few explicit solutions for conditions where the thermal conductivity or diffusion tensors are anisotropic. Such solutions have some use in making predictions for idealization of real systems, but are perhaps most useful for providing benchmark solutions which can be used to validate numerical codes. In this short paper, we present the transient solution to the diffusion equation in a cube under conditions of orthotropic anisotropy in the effective thermal conductivity or diffusion tensor. In particular, we consider the physically-relevant case of transport in a cube with no-flux boundaries for several initial conditions including: (1) a delta function, (2) a truncated Gaussian function, (3) a step function, and (4) a planar function. The potential relevance for each of these initial conditions in the context of validating numerical codes is discussed.
We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion-reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. W
In this paper, we consider a drift-diffusion system describing the corrosion of an iron based alloy in a nuclear waste repository. In comparison with the classical drift-diffusion system arising in the modeling of semiconductor devices, the originali
The large time behavior of zero mass solutions to the Cauchy problem for a convection-diffusion equation. We provide conditions on the size and shape of the initial datum such that the large time asymptotics of solutions is given either by the deriva
In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ partial_t u+(-Delta)^{frac{theta}{2}}u=0quadmbox{in}quad{bf R}^Ntimes(0,infty), qquad u(x,0)=varphi(x)quadmbox{in}quad{bf R}^N, $$ where $0<theta<2$
This article is concerned with the existence and the long time behavior of weak solutions to certain coupled systems of fourth-order degenerate parabolic equations of gradient flow type. The underlying metric is a Wasserstein-like transportation dist