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12 pages, 313 KiB

Open AccessArticle

Diameter Estimate in Geometric Flows

by Shouwen Fang and Tao Zheng

Mathematics 2023, 11(22), 4659; https://doi.org/10.3390/math11224659 - 16 Nov 2023

Viewed by 947

Abstract

We prove the upper and lower bounds of the diameter of a compact manifold

(M, g (t))

with

{dim}_{R} M = n (n \geq 3)

and a family of Riemannian metrics

g (t)

[...] Read more.

We prove the upper and lower bounds of the diameter of a compact manifold

(M, g (t))

with

{dim}_{R} M = n (n \geq 3)

and a family of Riemannian metrics

g (t)

satisfying some geometric flows. Except for Ricci flow, these flows include List–Ricci flow, harmonic-Ricci flow, and Lorentzian mean curvature flow on an ambient Lorentzian manifold with non-negative sectional curvature. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

22 pages, 396 KiB

Open AccessArticle

Determining the Coefficients of the Thermoelastic System from Boundary Information

by Xiaoming Tan

Mathematics 2023, 11(9), 2147; https://doi.org/10.3390/math11092147 - 4 May 2023

Viewed by 1397

Abstract

Given a compact Riemannian manifold

(M, g)

with smooth boundary

\partial M

, we give an explicit expression for the full symbol of the thermoelastic Dirichlet-to-Neumann map

Λ_{g}

with variable coefficients [...] Read more.

Given a compact Riemannian manifold

(M, g)

with smooth boundary

\partial M

, we give an explicit expression for the full symbol of the thermoelastic Dirichlet-to-Neumann map

Λ_{g}

with variable coefficients

λ, μ, α, β \in C^{\infty} (\bar{M})

. We prove that

Λ_{g}

uniquely determines partial derivatives of all orders of these coefficients on the boundary

\partial M

. Moreover, for a nonempty smooth open subset

Γ \subset \partial M

, suppose that the manifold and these coefficients are real analytic up to

Γ

. We show that

Λ_{g}

uniquely determines these coefficients on the whole manifold

\bar{M}

. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

13 pages, 290 KiB

Open AccessArticle

Large-Time Behavior of Momentum Density Support of a Family of Weakly Dissipative Peakon Equations with Higher-Order Nonlinearity

by Xianxian Su and Xiaofang Dong

Mathematics 2023, 11(6), 1325; https://doi.org/10.3390/math11061325 - 9 Mar 2023

Cited by 1 | Viewed by 1269

Abstract

In this paper, we mainly study the weakly dissipative peakon equations with higher-order nonlinearity. Under the effect of dissipation, we first derive the infinite propagation speed if the initial datum has a nonnegative compact support. Furthermore, we obtain the large-time behavior of the [...] Read more.

In this paper, we mainly study the weakly dissipative peakon equations with higher-order nonlinearity. Under the effect of dissipation, we first derive the infinite propagation speed if the initial datum has a nonnegative compact support. Furthermore, we obtain the large-time behavior of the support of momentum density with the initial data compactly supported. The corresponding results are obtained by using some prior estimates and the energy method. It is worth noting that we need to overcome the difficulties caused by the high-order nonlinear structure and the dissipative effect of the equation. The obtained results generalize the previous results to a certain degree. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

23 pages, 361 KiB

Open AccessArticle

Quasilinear Parabolic Equations Associated with Semilinear Parabolic Equations

by Katsuyuki Ishii, Michel Pierre and Takashi Suzuki

Mathematics 2023, 11(3), 758; https://doi.org/10.3390/math11030758 - 2 Feb 2023

Viewed by 1876

Abstract

We formulate a quasilinear parabolic equation describing the behavior of the global-in-time solution to a semilinear parabolic equation. We study this equation in accordance with the blow-up and quenching patterns of the solution to the original semilinear parabolic equation. This quasilinear equation is [...] Read more.

We formulate a quasilinear parabolic equation describing the behavior of the global-in-time solution to a semilinear parabolic equation. We study this equation in accordance with the blow-up and quenching patterns of the solution to the original semilinear parabolic equation. This quasilinear equation is new in the theory of partial differential equations and presents several difficulties for mathematical analysis. Two approaches are examined: functional analysis and a viscosity solution. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

11 pages, 264 KiB

Open AccessArticle

The Buckling Operator: Inverse Boundary Value Problem

by Yanjun Ma

Mathematics 2023, 11(2), 268; https://doi.org/10.3390/math11020268 - 4 Jan 2023

Viewed by 1187

Abstract

In this paper, we consider a zeroth-order perturbation

q (x)

of the buckling operator

Δ^{2} - κ Δ

, which can be uniquely determined by measuring the Dirichlet-to-Neumann data on the boundary. We extend the conclusion of the biharmonic operator [...] Read more.

In this paper, we consider a zeroth-order perturbation

q (x)

of the buckling operator

Δ^{2} - κ Δ

, which can be uniquely determined by measuring the Dirichlet-to-Neumann data on the boundary. We extend the conclusion of the biharmonic operator to the buckling operator, but the Dirichlet-to-Neumann map given in this study is more meaningful and general. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

10 pages, 257 KiB

Open AccessArticle

Liouville-Type Theorem for Nonlinear Elliptic Equations Involving Generalized Greiner Operator

by Wei Shi

Mathematics 2023, 11(1), 61; https://doi.org/10.3390/math11010061 - 24 Dec 2022

Cited by 1 | Viewed by 1359

Abstract

In this paper, we study the Liouville-type behaviors of the generalized Greiner operators with nonlinear boundary value conditions. We use the technique based upon the method of moving planes. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

15 pages, 318 KiB

Open AccessArticle

Positive Radially Symmetric Entire Solutions of p-k-Hessian Equations and Systems

by Wei Fan, Limei Dai and Bo Wang

Mathematics 2022, 10(18), 3258; https://doi.org/10.3390/math10183258 - 7 Sep 2022

Cited by 6 | Viewed by 1813

Abstract

In this paper, we discuss the existence of positive radially symmetric entire solutions of the p-k-Hessian equation [...] Read more.

In this paper, we discuss the existence of positive radially symmetric entire solutions of the p-k-Hessian equation

σ_{k}^{\frac{1}{k}} (λ (D_{i} ({| D u |}^{p - 2} D_{j} u))) = α^{\frac{1}{k}} (| x |) f (u),

and the general p-k-Hessian system

σ_{k}^{\frac{1}{k}} (λ (D_{i} ({| D u |}^{p - 2} D_{j} u))) = α^{\frac{1}{k}} (| x |) f_{1} (v) f_{2} (u)

,

σ_{k}^{\frac{1}{k}} (λ (D_{i} ({| D v |}^{p - 2} D_{j} v))) = β^{\frac{1}{k}} (| x |) g_{1} (u) g_{2} (v)

. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

16 pages, 294 KiB

Open AccessArticle

Parabolic Hessian Equations Outside a Cylinder

by Limei Dai and Xuewen Guo

Mathematics 2022, 10(16), 2839; https://doi.org/10.3390/math10162839 - 9 Aug 2022

Cited by 1 | Viewed by 1336

Abstract

In this article, we mainly review the parabolic Hessian equation on the exterior region. The existence and uniqueness of solutions with asymptotic properties to the exterior problem of the parabolic Hessian equation were obtained by using the Perron method. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

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