Some gradient estimates for nonlinear heat-type equations on smooth metric measure spaces with compact boundary

In this paper we prove some Hamilton type and Li-Yau type gradient estimates on positive solutions to generalized nonlinear parabolic equations on smooth metric measure space with compact boundary. The geometry of the space in terms of lower bounds on the weighted Bakry-Emery Ricci curvature tensor and weighted mean curvature of the boundary are key in proving generalized local and global gradient estimates. Various applications of these gradient estimates in terms of parabolic Harnack inequalities and Liouville type results are discussed. Further consequences to some specific models informed by the nature of the nonlinearities are highlighted.Comment: 4

Analysis of eigenvalues and conjugate heat kernel under the Ricci flow

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Generalized parabolic frequency on compact manifolds

In this paper, we first prove monotonicity of a generalized para bolic frequency on weighted closed Riemannian manifolds for some linear heat equation. Secondly, a certain generalized parabolic frequency functional is defined with respect to the solutions of a nonlinear weighted p-heat-type equation on manifolds, and its monotonicity is proved. Notably, the monotonicities are derived with no assumption on both the curvature and the potential function. Further consequences of these monotonicity formulas from which we can get backward uniqueness are discussedComment: 14 page

Parabolic frequency monotonicity on the conformal Ricci flow

This paper is devoted to the investigation of the monotonicity of parabolic frequency functional under conformal Ricci flow defined on a closed Riemannian manifold of constant scalar curvature and dimension not less than 3. Parabolic frequency functional for solutions of certain linear heat equation coupled with conformal pressure is defined and its monotonicity under the conformal Ricci flow is proved by applying Bakry-Emery Ricci curvature bounds. Some consequences of the monotonicity are also presented.Comment: 18 page

GRADIENT ESTIMATES FOR A NONLINEAR ELLIPTIC EQUATIONON COMPLETE NONCOMPACT RIEMANNIAN MANIFOLD

Let(M,g)be ann-dimensional complete noncompact Riemannian manifold (withpossibly empty boundary). We derive local and global gradient estimates on positive solutionsu(x)to the following nonlinear elliptic equationΔu(x)+aus(x)+λ(x)u(x)=0,x∈M,whereaandsare constants,a∈R\{0},s>1andλ(x)is bounded onM. Our gradientestimates yield differential Harnack inequalities as an application. This paper extends results ofY. Yang [17]andJ.Li[11, Theorem 3.1]

Geometric estimates on weighted p-fundamental tone and applications to the first eigenvalue of submanifolds with bounded mean curvature

This paper generalizes to the context of smooth metric measure spaces and submanifolds with negative sectional curvatures some well-known geometric estimates on the p-fundamental tone by using vector fields satisfying a positive divergence condition. Choosing the vector field to be the gradient of an appropriately chosen distance function yields generalised McKean estimates whilst other choices of vector fields yield new geometric estimates generalising certain results of Lima et al. (Nonlinear Anal. 2010;72:771–781). We also obtain a lower bound on the spectrum of the weighted p-Laplacian on a complete noncompact smooth metric space with the underlying space being a submanifold with bounded mean curvature in the hyperbolic space form of constant negative sectional curvature generalising results of Du and Mao (J Math Anal Appl. 2017;456:787–795)

Elliptic gradient estimates for a nonlinear f-heat equation on weighted manifolds with evolving metrics and potentials

We develop local elliptic gradient estimates for a basic nonlinear f-heat equation with a logarithmic power nonlinearity and establish pointwise upper bounds on the weighted heat kernel, all in the context of weighted manifolds, where the metric and potential evolve under a Perelman-Ricci type flow. For the heat bounds use is made of entropy monotonicity arguments and ultracontractivity estimates with the bounds expressed in terms of the optimal constant in the logarithmic Sobolev inequality. Some interesting consequences of these estimates are presented and discussed

Renal Artery Aneurysm at a Nigerian Tertiary Centre: Case Report and Review of Literature

Renal artery aneurysms are rare urologic conditions, with rupture being the most feared complication. We discuss the management of two womenwith this disease at our center. The first was a 58‑year‑old woman who presented with torrential hematuria and hemodynamic compromise.  Abdominal computed tomography (CT) angiography revealed a left renal artery aneurysm, and she had emergency nephrectomy. The second was a 40‑year‑old woman with recurrent flank pain of 2 years duration. Serial CT scans showed a calcified renal aneurysm remaining stable over this period. She was managed nonoperatively, with serial follow‑up imaging to determine if future intervention is warranted. We conclude on the need for adequate evaluation and imaging to promptly diagnose renal artery aneurysms, and that care should be individualized. Keywords: Aneurysm, angiography, artery, nephrectomy, renal, ruptur