In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving q(x)-Laplacian parabolic equation . The potential is not necessarily smooth but belongs to a Sobolev space . Given the initial datum as a probability density on , we use a descent algorithm in the probability space to discretize the q(x)-Laplacian parabolic equation in time. Then, we use compact embedding ↪↪ established by Fan and Zhao to study the convergence of our algorithm to a weak solution of the q(x)-Laplacian parabolic equation. Finally, we establish the convergence of solutions of the q(x)-Laplacian parabolic equation to equilibrium in the p(.)-variable exponent Wasserstein space.

中文翻译：

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演化q（x）-Laplacian方程正解的存在性与渐近行为

在本文中，我们将M. Agueh的变分方法扩展到涉及q（x）-Laplacian抛物方程的一类抛物方程。势不一定是平滑的，而是属于Sobolev空间。给定初始基准作为on上的概率密度，我们在概率空间中使用下降算法来及时离散q（ x）-Laplacian抛物方程。然后，我们用紧嵌入↪↪范，赵建立我们的算法的收敛学习到的弱解q（ X）-Laplacian抛物线方程。最后，我们建立q（ x）-Laplacian抛物方程到p处平衡的解的收敛性。（。）-变量指数Wasserstein空间。