Journal of Differential Equations最新文献

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Continuity of attractors of parabolic equations with nonlinear boundary conditions and rapidly varying boundaries. The case of a Lipschitz deformation

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-02-21 DOI: 10.1016/j.jde.2025.02.041

Gleiciane S. Aragão , José M. Arrieta , Simone M. Bruschi

In this paper we obtain the continuity of attractors for nonlinear parabolic equations with nonlinear boundary conditions when the boundary of the domain varies rapidly as a parameter ϵ goes to zero. We consider the case where the boundary of the domain presents a highly oscillatory behavior as the parameter ϵ goes to zero. For the case where we have a Lipschitz deformation of the boundary with the Lipschitz constant uniformly bounded in ϵ but the boundaries do not approach in a Lipschitz sense, the solutions of these equations converge in certain sense to the solution of a limit parabolic equation of the same type but where the boundary condition has a factor that captures the oscillations of the boundary. To address this problem, it is necessary to consider the notion of convergence of functions defined in varying domains and the convergence of a family of operators defined in different Banach spaces. Moreover, since we consider problems with nonlinear boundary conditions, it is necessary to extend these concepts to the case of spaces with negative exponents and to operators defined between these spaces.

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引用次数: 0

Recovering initial states in semilinear parabolic problems from time-averages

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-02-21 DOI: 10.1016/j.jde.2025.02.049

Lina Sophie Schmitz, Christoph Walker

Well-posedness of certain semilinear parabolic problems with nonlocal initial conditions is shown in time-weighted spaces. The result is applied to recover the initial states in semilinear parabolic problems with nonlinearities of superlinear behavior near zero from small time-averages over arbitrary time periods.

引用次数: 0

Polyhedral entire solutions in reaction-diffusion equations

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-02-21 DOI: 10.1016/j.jde.2025.02.034

Masaharu Taniguchi

This paper studies polyhedral entire solutions to a bistable reaction-diffusion equation in

R^{n}

. We consider a pyramidal traveling front solution to the same equation in

R^{n + 1}

. As the speed goes to infinity, its projection converges to an n-dimensional polyhedral entire solution. Conversely, as the time goes to −∞, an n-dimensional polyhedral entire solution gives n-dimensional pyramidal traveling front solutions. The result in this paper suggests a correlation between traveling front solutions and entire solutions in general reaction-diffusion equations or systems.

引用次数: 0

Dynamic boundary flux-driven shallow waters: Insights from a dissipative-dispersive system

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-02-21 DOI: 10.1016/j.jde.2025.02.055

Neng Zhu , Kun Zhao

<div><div>This paper is concerned with a shallow water system under dynamic boundary conditions:<span><span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mo>(</mo><mi>u</mi><mi>w</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mi>ϵ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>,</mo><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><mi>w</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mi>μ</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>δ</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>x</mi><mi>x</mi><mi>t</mi></mrow></msub><mo>,</mo><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>α</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>β</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>w</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>w</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>.</mo></math></span></span></span> By constructing suitable relative entropy functionals, it is shown that under certain conditions on <span><math><mi>α</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and <span><math><mi>β</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>, classical solutions with potentially large energy exist globally in time, and the solutions converge to the equilibria determined by the initial and boundary conditions. The results hold for all values of <span><math><mi>m</mi><mo>⩾</mo><mn>1</mn></math></span> when <span><math><mi>μ</mi><mo>></mo><mn>0</mn></math></span>, and for <span><math><mi>m</mi><mo>⩾<

引用次数: 0

Nondegeneracy of positive solutions for a biharmonic Hartree equation and its application

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-02-20 DOI: 10.1016/j.jde.2025.02.024

Minbo Yang , Weiwei Ye , Xinyun Zhang

We study the nondegeneracy of positive solutions of the following biharmonic Hartree equation

Δ^{2} u = (| x |^{- α} ⁎ | u |^{p}) u^{p - 1}, in R^{N},

where

p = \frac{2 N - α}{N - 4}

N \geq 5

and

0 < α < N

. Our method relies on the spherical harmonic decomposition and the Funk-Heck formula of the spherical harmonic functions. Then as an application, by applying a finite dimension reduction and local Pohožaev identity, we can construct multi-bubble solutions for the following equation with potential

Δ^{2} u + V (| x^{'} |, x^{″}) u = (| x |^{- α} ⁎ | u |^{p}) u^{p - 1} x \in R^{N},

where

N \geq 9

(x^{'}, x^{″}) \in R^{2} \times R^{N - 2}

and

V (| x^{'} |, x^{″})

is a bounded and nonnegative function. We prove that the existence result is restricted to the range

6 - \frac{12}{N - 4} \leq α < N

which shows the influence of the order of Riesz potential.

引用次数: 0

A uniform C1 connecting lemma for singular flows

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-02-19 DOI: 10.1016/j.jde.2025.02.043

Ming Li , Xingzhong Liu

We extend the uniform

C^{1}

connecting lemma [28] to singular flows. The perturbation parameters are valid in a

C^{1}

-neighborhood of the given vector field. Moreover, they are also uniform on the points except singularities and some periodic ones with small period.

引用次数: 0

Gradient estimate for solutions to Δbu = 0 and Δbu+auln⁡u+bu=0 on pseudohermitian manifold

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-02-19 DOI: 10.1016/j.jde.2025.02.045

Shitong Niu, Jun Sun

In this paper, we will establish fundamental

C^{1}

estimate for solutions to

Δ_{b} u = 0

and

C^{0}

and

C^{1}

estimates for solutions to a class of nonlinear elliptic equations of the form

Δ_{b} u + a u \ln u + b u = 0

on pseudohermitian manifold. The proof consists of combining the Nash-Moser iteration and Saloff-Coste type Sobolev inequalities.

引用次数: 0

New type of solutions for the critical Lane-Emden system

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-02-19 DOI: 10.1016/j.jde.2025.02.046

Wenjing Chen, Xiaomeng Huang

In this paper, we consider the critical Lane-Emden system

{\begin{matrix} - Δ u = K_{1} (y) v^{p}, y \in R^{N}, \\ - Δ v = K_{2} (y) u^{q}, y \in R^{N}, \\ u, v > 0, \end{matrix}

where

N \geq 5

p, q \in (1, \infty)

with

\frac{1}{p + 1} + \frac{1}{q + 1} = \frac{N - 2}{N}

K_{1} (y)

and

K_{2} (y)

are positive radial potentials. Under suitable conditions on

K_{1} (y)

and

K_{2} (y)

, we construct a new family of solutions to this system, which are centred at points lying on the top and the bottom circles of a cylinder.

引用次数: 0

Quasilinear parabolic equations with superlinear nonlinearities in critical spaces

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-02-19 DOI: 10.1016/j.jde.2025.02.039

Bogdan-Vasile Matioc , Luigi Roberti , Christoph Walker

Well-posedness in time-weighted spaces for quasilinear (and semilinear) parabolic evolution equations

u^{'} = A (u) u + f (u)

is established in a certain critical case of strict inclusion

dom (f) ⊊ dom (A)

for the domains of the (superlinear) function

u \mapsto f (u)

and the quasilinear part

u \mapsto A (u)

. Based upon regularizing effects of parabolic equations, it is proven that the solution map generates a semiflow in a critical intermediate space. The applicability of the abstract results is demonstrated by several examples including a model for atmospheric flows and semilinear and quasilinear evolution equations with scaling invariance for which well-posedness in the critical scaling invariant intermediate spaces is shown.

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引用次数: 0

Dispersive decay for the energy-critical nonlinear Schrödinger equation

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-02-19 DOI: 10.1016/j.jde.2025.02.040

Matthew Kowalski

We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear Schrödinger equation in spatial dimensions

d = 3, 4

for both the initial-value and final-state problems.

引用次数: 0

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