In this paper, the authors study the multiplicity of solutions to the weighted p-Laplacian with isolated singularity and diffusion suppressed by convection $ - {\rm{div}}\left( {{{\left| x \right|}^\alpha }{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) + \lambda {1 \over {{{\left| x \right|}^\beta }}}{\left| {\nabla u} \right|^{p - 2}}\nabla u \cdot x = {\left| x \right|^\gamma }g\left( {\left| x \right|} \right)\,\,\,\,{\rm{in}}\,\,B\backslash \left\{ 0 \right\}$ subject to nonlinear Robin boundary value condition ${\left| x \right|^\alpha }{\left| {\nabla u} \right|^{p - 2}}\nabla u \cdot \vec n = A - \rho u\,\,\,{\rm{on}}\,\,\partial B,$ where λ > 0, B ⊂ ℝ ^N (N ≥ 2) is the unit ball centered at the origin, α > 0, p > 1, β ∈ ℝ, γ > −N, g ∈ C([0, 1]) with g(0) > 0, A ∈ ℝ, ρ > 0 and ${\vec n}$ is the unit outward normal. The same problem with diffusion promoted by convection, namely λ ≤ 0, has already been discussed by the last two authors (Song-Yin (2012)), where the existence, nonexistence and classification of singularities for solutions are presented. Completely different from [Song, H. J. and Yin, J. X., Removable isolated singularities of solutions to the weighted p-Laplacian with singular convection, Math. Meth. Appl. Sci., 35, 2012, 1089–1100], in the present case λ > 0, namely the diffusion is suppressed by the convection, non-singular solutions are not only existent but also may be infinite which vary according only to the values of solutions at the isolated singular point. At the same time, the singular solutions may exist only if the diffusion dominates the convection.

In this paper, the authors consider the asymptotic synchronization of a linear dissipative system with multiple feedback dampings. They first show that under the observability of a scalar equation, Kalman’s rank condition is sufficient for the uniqueness of solution to a complex system of elliptic equations with mixed observations. The authors then establish a general theory on the asymptotic stability and the asymptotic synchronization for the corresponding evolutional system subjected to mixed dampings of various natures. Some classic models are presented to illustrate the field of applications of the abstract theory.

In this paper, the authors study the asymptotically linear elliptic equation on manifold with conical singularities $ - {\Delta _{\mathbb{B}}}u + \lambda u = a\left( z \right)f\left( u \right),\,\,\,\,\,\,u \ge 0\,\,{\rm{in}}\,\,_ + ^N,$

where N = n + 1 ≥ 3, λ > 0, z = (t, x ₁, ⋯, x _n), and ${\Delta _{\mathbb{B}}} = {\left( {t{\partial _t}} \right)^2} + \partial _{{x_1}}^2 + \cdots + \partial _{{x_n}}^2$. Combining properties of cone-degenerate operator, the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation, we obtain a positive solution under some suitable conditions on a and f.

In this paper, the authors will apply De Giorgi-Nash-Moser iteration to establish boundary Hölder estimates for a class of degenerate elliptic equations in piecewise C ²-smooth domains.

A complete manifold is said to be nonparabolic if it does admit a positive Green’s function. To find a sharp geometric criterion for the parabolicity/nonparbolicity is an attractive question inside the function theory on Riemannian manifolds. This paper devotes to proving a criterion for nonparabolicity of a complete manifold weakened by the Ricci curvature. For this purpose, we shall apply the new Laplacian comparison theorem established by the first author to show the existence of a non-constant bounded subharmonic function.

Given initial data u ₀ ∈ L ^p (ℝ³) for some p in $\left[ {3,{{18} \over 5}} \right[$, the auhtors first prove that 3D incompressible Navier-Stokes system has a unique solution u = u _L+v with ${u_L}\mathop = \limits^{{\rm{def}}} \,{{\rm{e}}^{t\Delta }}{u_0}$ and $v \in {{\tilde L}^\infty }\left( {\left[ {0,T} \right];{{\dot H}^{{5 \over 2} - {6 \over p}}}} \right) \cap {{\tilde L}^1}\left( {\left] {0,T} \right[;{{\dot H}^{{9 \over 2} - {6 \over p}}}} \right)$ for some positive time T. Then they derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in [Chemin, J.-Y., Gallagher, I. and Zhang, P., On the radius of analyticity of solutions to semi-linear parabolic system, Math. Res. Lett., 27, 2020, 1631–1643, Herbst, I. and Skibsted, E., Analyticity estimates for the Navier-Stokes equations, Adv. in Math., 228, 2011, 1990–2033] with initial data in Ḣ ^s(ℝ³) for $s \in \left[ {{1 \over 2},{3 \over 2}} \right[$.

In this paper, the authors show that one cannot dream of the positivity of the Hayward energy in the general situation. They consider a scenario of a spherically symmetric constant density star matched to the Schwarzschild solution, representing momentarily static initial data. It is proved that any topological tori within the star, distorted or not, have strictly positive Hayward energy. Surprisingly we find analytic examples of ‘thin’ tori with negative Hayward energy in the outer neighborhood of the Schwarzschild horizon. These tori are swept out by rotating the standard round circles in the static coordinates but they are distorted in the isotropic coordinates. Numerical results also indicate that there exist horizontally dragged tori with strictly negative Hayward energy in the region between the boundary of the star and the Schwarzschild horizon.

The author gives an alternative and simple proof of the global existence of smooth solutions to the Cauchy problem for wave maps from the (1+2)-dimensional Minkowski space to an arbitrary compact smooth Riemannian manifold without boundary, for arbitrary smooth, radially symmetric data. The author can also treat non-compact manifold under some additional assumptions which generalize the existing ones.

The Faddeev model is a fundamental model in relativistic quantum field theory used to model elementary particles. The Faddeev model can be regarded as a system of non-linear wave equations with both quasi-linear and semi-linear non-linearities, which is particularly challenging in two space dimensions. A key feature of the system is that there exist undifferentiated wave components in the non-linearities, which somehow causes extra difficulties. Nevertheless, the Cauchy problem in two space dimenions was tackled by Lei-Lin-Zhou (2011) with small, regular, and compactly supported initial data, using Klainerman’s vector field method enhanced by a novel angular-radial anisotropic technique. In the present paper, the authors revisit the Faddeev model and remove the compactness assumptions on the initial data by Lei-Lin-Zhou (2011). The proof relies on an improved L ² norm estimate of the wave components in Theorem 3.1 and a decomposition technique for non-linearities of divergence form.

In the present article, the authors find and establish stability of multiplier ideal sheaves, which is more general than strong openness.

The Darboux transformation for the two dimensional A _2n−1 ⁽²⁾ Toda equations is constructed so that it preserves all the symmetries of the corresponding Lax pair. The expression of exact solutions of the equation is obtained by using Darboux transformation.

A Hermitian curvature flow on a compact Calabi-Yau manifold is proposed and a regularity result is obtained. The solution of the flow, if exists, is a balanced Hermitian-Einstein metric.

Aircraft comes out at the beginning of the last century. Accompanied by the progress of high speed flight the theory of partial differential equations has been greatly developed. This paper gives a brief review on the history of applications of partial differential equations to the study of supersonic flows arising in high speed flight.

In this paper, the authors derive Hölder gradient estimates for graphic functions of minimal graphs of arbitrary codimensions over bounded open sets of Euclidean space under some suitable conditions.

In this paper, the authors give a survey about λ-hypersurfaces in Euclidean spaces. Especially, they focus on examples and rigidity of λ-hypersurfaces in Euclidean spaces.