In this talk I report my joint work with Hui-Chun Zhang to give a complete answer to the question. We show that every weak solution of the harmonic map heat flow into CAT(0) spaces is Lipschitz continuous in both space and time. Very recently, Lin and Wang gave an alternative proof.

Let $M=M_1\times M_2$ be a product of geodesically complete Riemannian manifolds, and let $p_1,p_2\in(1,\infty)$. We introduce a mixed-norm capacity associated with the Green operator on $M$ and the mixed Lebesgue space $L^{p_2}(L^{p_1})(M)$. This capacity leads to notions of $L^{p_2}(L^{p_1})$-parabolicity and the $L^{p_2}(L^{p_1})$-Liouville property for product manifolds, where $1/p_i+1/p_i'=1$ for $i=1,2$. We establish equivalent characterizations of this mixed capacity and prove the existence of capacitary measures represented by a nonlinear mixed potential. Our main result is a mixed-norm analogue of the equivalence between parabolicity, Green-kernel integrability, and the Liouville property: $M$ is $L^{p_2}(L^{p_1})$-parabolic if and only if, for some, equivalently for every, $x\in M$, the restriction $G^M(x;\cdot)\mathbf{1}_{M\setminus B(x,r)}$ does not belong to $L^{p_2'}(L^{p_1'})(M)$; this is further equivalent to the $L^{p_2'}(L^{p_1'})$-Liouville property. Under a radial Harnack-type inequality—for example, under Li–Yau heat kernel estimates, and hence for products with nonnegative Ricci curvature—these conditions are also equivalent to the divergence of the nonlinear mixed potential $\mathcal{G}_{p_1,p_2}(f)$ for every nonzero, nonnegative $f\in C_c^\infty(M)$. Finally, we derive explicit volume-growth criteria and apply them to Euclidean products, obtaining a sharp mixed-norm Liouville criterion for $\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$.

In this talk, I will discuss classification problems inspired by this line of inquiry, but extended to entire solutions of the parabolic Allen-Cahn equation. Specifically, I will present a recent joint work with my current PhD student, Lubo Wang, demonstrating that if the blow-down limit of an entire solution is a shrinking sphere, then the solution is radially symmetric in the spatial directions.

We will talk about the following boundary value problem of Monge-Ampère equation. Under suitable conditions, we will show the problem is solvable. This is a joint work with Weiming Shen.

In this talk, we give some recent results on the three dimensional logarithmic Schrödinger-Poisson system with coercive potential. Under some technical conditions on the potential, by using variational methods we investigate the existence, asymptotic behavior and radial symmetry of ground state.