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Abstract

In this paper, the initial value problem of the convection-diffusion equation of Burgers type is treated. In the asymptotic profile of solutions, the nonlinearity of the equation is reflected. A characteristic component derived from nonlinearity develops logarithmically over time, and the rate of decay varies depending on the spatial dimensions. The rate of this component predicted from the scale of the equation. In even dimensions, the logarithmic component decays at the expected rate, but in odd dimensions, it decays much faster than expected. These results suggest that there are differences in the symmetry of nonlinearity depending on the parity of dimensions. This interpretation is supported by comparison with similar Navier--Stokes equations. The Burgers type is applicable as an indicator for considering several bilinear problems.