We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space \(\mathbb {R}^n\), where \(n \ge 2\), assuming that the diffusion matrix depends on the space variable x and has a finite limit along any ray as \(|x| \rightarrow \infty \). Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose profile is entirely determined by the asymptotic diffusion matrix. Examples are given which show that the profile can be a rather general Gaussian-like function, and that the approach to the self-similar solution can be arbitrarily slow depending on the continuity and coercivity properties of the asymptotic matrix. The proof of our results relies on appropriate energy estimates for the diffusion equation in self-similar variables. The new ingredient consists in estimating not only the difference w between the solution and the self-similar profile, but also an antiderivative W obtained by solving a linear elliptic problem which involves w as a source term. Hence, a good part of our analysis is devoted to the study of linear elliptic equations whose coefficients are homogeneous of degree zero.

我们假设线性矩阵或半线性抛物方程在整个空间\（\ mathbb {R} ^ n \）中，其中\（n \ ge 2 \）的局部解的长期行为，假设扩散矩阵取决于空间变量x，并且沿任何射线都有一个有限极限，例如\（| x | \ rightarrow \ infty \）。在非线性情况下的适当小条件下，我们证明了它的轮廓完全由渐近扩散矩阵确定的自相似解的收敛性。给出的例子表明，轮廓可以是一个相当一般的高斯函数，并且根据渐近矩阵的连续性和矫顽性，自相似解的方法可以任意慢。我们的结果证明依赖于自相似变量中扩散方程的适当能量估计。新成分不仅在于估计溶液和自相似曲线之间的差异w，而且还包括通过求解涉及w的线性椭圆问题获得的反导数W。作为源术语。因此，我们的分析工作的大部分致力于研究系数为零次齐次的线性椭圆方程。