Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations

We consider the Hardy–Hénon parabolic equation ${u_t-\Delta u =|x|^a |u|^{p-1}u}$ with p > 1 and ${a\in \mathbb{R}}$ . We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial-boundary value problem.     

Vanishing theorems for some classes of Riemann–Cartan manifolds

In this paper, a classification of Riemann–Cartan manifolds based on the orthogonal decomposition of the torsion tensor is given. Problems on the existence of two classes ℘₁ ⨁ ℘₂ and ℘₃ of Riemann–Cartan spaces are discussed.     

Liouville-type results for solutions of −Δu=|u|p−1u on unbounded domains of RN

In this Note we study solutions, possibly unbounded and sign-changing, of the equation $?\mathrm{\Delta }u={|u|}^{p?1}u$ on unbounded domains of ${\mathbb{R}}^N$ with $N?2$ and $p>1$. We prove some Liouville-type results and a classification theorem for $C^2$ solutions belonging to one of the following classes: stable solutions, finite Morse index solutions and solutions which are stable outside a compact set. We also extend, to smooth coercive epigraphs, the well-known results of Gidas and Spruck concerning non-negative solutions of the considered equation. To cite this article: A. Farina, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Spectral estimates for Schrödinger and Dirac-type operators on Riemannian manifolds

Supported by funds of M.U.R.S.T. (Italy). The author is grateful to S. Gallot for his encouragement and for helpful discussions and to G. Besson for some interesting remarks     

Comparison and vanishing theorems for Kähler manifolds

In this paper, we consider orthogonal Ricci curvature \(Ric^{\perp }\) for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate subtle relationship among them. As a consequence of the vanishing theorem, we show that any compact Kähler manifold with positive orthogonal Ricci curvature must be projective. This result complements a recent result of Yang (RC-positivity, rational connectedness, and Yau’s conjecture. arXiv:1708.06713) on the projectivity under the positivity of holomorphic sectional curvature. The simply-connectedness is shown when the complex dimension is smaller than five. Further study of compact Kähler manifolds with \(Ric^{\perp }>0\) is carried in Ni et al. (Manifolds with positive orthogonal Ricci curvature. arXiv:1806.10233).     

Plünnecke and Kneser type theorems for dimension estimates

Given a division ring K containing the field k in its center and two finite subsets A and B of K*, we give some analogues of Plünnecke and Kneser Theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. We also explain how they imply the corresponding more classical theorems for abelian groups. These Plünnecke type estimates are then generalized to the case of associative algebras. We also obtain an analogue in the context of division rings of a theorem by Tao describing the sets of small doubling in a group.     

Multilinear estimates for the Laplace spectral projectors on compact manifolds

The purpose of this Note is to extend to any space dimension the bilinear estimate for eigenfunctions of the Laplace operator on a compact manifold (without boundary) obtained by the authors (preprint: http://www.arxiv.org/abs/math/0308214) in dimension 2. We also give some related trilinear estimates. To cite this article: N. Burq et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

An improperly posed estimate for the equation |u|α ut =uxx +F(u)(0 <α <1)

The noncharacteristic Cauchy problem of heat equations is not well-posed. But the estimation on the continuous dependence of solutions holds under their prescribed bound and the bound of their Cauchy data. In this paper we show that a similar estimation holds also for some degenerate quasilinear parabolic equation.     

Eigenvalue estimates for the Dirac operator on quaternionic K?hler manifolds

We consider the Dirac operator on compact quaternionic K?hler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space. Received April 21, 1998; in final form June 16, 1998     

Comparison theorems for nonlinear evolution equations and their application to the investigation of the dependence of the solutions of the equation ut=δϕ(u) ON ϕ and on the boundary conditions

One obtains estimates of the form, whereu. are generalized solutions of the equationsdu/dt=Au, du/dt=Bu whileA, B are non-linear,m-dissipative operators in a Banach space, and there exists an operatorP:D(A)D(B), such thatPw · W+BPw –Aw, uniformly on some setw. These results are applied to the investigation of the dependence of the solutions of the Cauchy, Dirichlet problems and of the problem with the boundary condition –du/dn=(u) for the equation u₁=(u) on the continuous nondecreasing functions and.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 183–198, 1984.The author is sincerely grateful to O. A. Ladyzhenskaya and N. N. Ural'tseva for their interest in this paper and for useful discussions.     

A note on the problem −Δu=λu+u|u|2*−2

The dual approach to the problem –u=u+u|u|^2*–2, uþe ₀ ¹ (gW), permits a simple proof of a recent existence result [5] and allows extensions of this result to similar problems also with asymmetric mnonlinearities.Supported by min P.I., Gruppo Naz. (40%) Calcolo delle Variazioni...