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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 distance for vector-valued functions, with nonlinear mobilities in each component. Under the hypothesis of (flat) convexity of the driving free energy functional, weak solutions are constructed by means of the variational minimizing movement scheme for metric gradient flows. The essential regularity estimates are derived by variational methods.
A nonlinear parabolic equation of sixth order is analyzed. The equation arises as a reduction of a model from quantum statistical mechanics, and also as the gradient flow of a second-order information functional with respect to the $L^2$-Wasserstein
In this paper, the finite time extinction of solutions to the fast diffusion system $u_t=mathrm{div}(| abla u|^{p-2} abla u)+v^m$, $v_t=mathrm{div}(| abla v|^{q-2} abla v)+u^n$ is investigated, where $1<p,q<2$, $m,n>0$ and $Omegasubset mathbb{R}^N (N
This paper deals with the existence of positive solutions for the nonlinear system q(t)phi(p(t)u_{i}(t)))+f^{i}(t,textbf{u})=0,quad 0<t<1,quad i=1,2,...,n. This system often arises in the study of positive radial solutions of nonlinear elliptic syste
As an extension of our previous work in Sun et.al (2018) [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy functionals.
For a class of fourth order gradient flow problems, integration of the scalar auxiliary variable (SAV) time discretization with the penalty-free discontinuous Galerkin (DG) spatial discretization leads to SAV-DG schemes. These schemes are linear and