Inspired by the construction of blow-up solutions of the heat flow of harmonic maps from D^2 into S^2 via maximum principle (Chang et al., J. Diff. Geom., 36, 1992, pp. 507-515.) we provide examples of nonexistence of smooth axially symmetric harmonic maps from B^3 into S^2 with smooth boundary maps of degree zero.

In this paper we study well-posedness and asymptotic behavior of solution of a free boundary problem modeling the growth of multi-layer tumors under the action of an external inhibitor. We first prove that this problem is locally well-posed in little H“older spaces. Next we investigate asymptotic behavior of the solution. By making delicate analysis of spectrum of the linearization of the stationary free boundary problemand using the linearized stability theorem, we prove that if the surface tension coefficient γ is larger than γ^∗ > 0 the flat stationary solution is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficient small.

We study the homogeneous Dirichlet problem in a ball for semi-linear elliptic problems derived from the Brezis-Nirenberg one with concave-convex nonlinearities. We are interested in determining non-radial solutions which are invariant with respect to some subgroup of the orthogonal group. We prove that unlike separated nonlinearities, there are two types of solutions, one converging to zero and one diverging. We conclude at the end on the classification of non radial solutions related to the nonlinearity used.

In this work we derive local gradient estimates of the Aronson-Bénilan type for positive solutions of porous medium equations under Ricci flow with bounded Ricci curvature. As an application, we derive a Harnack type inequality.

We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain with the pressure law p(\varrho,ϑ)∼\varrhoϑ+\varrho ln^α(1+\varrho). For the heat flux q∼-(1+ϑ^m)∇ϑ we show the existence of a weak solution provided α > max{1,1/m}, m > 0. This improves the recent result from [1].

In this paper, we study the positive solutions for a class of integral systems and prove that all the solutions are radially symmetric and monotonically decreasing about some point. Moreover, we also obtain the uniqueness result for a special case. We use a new type of moving plane method introduced by Chen-Li-Ou [1]. Our new ingredient is the use of Hardy-Littlewood-Sobolev inequality instead of Maximum Principle.

Extending the previous work [1], we establish well-posedness results for a more general class of semilinear wave equations with exponential growth. First, we investigate the well-posedness in the energy space. Then, we prove the propagation of the regularity in the Sobolev spaces H^s(R^2) with s ≥ 1. Finally, an ill-posedness result is obtained in H^s(R^2) for s < 1.