Nonlinear Analysis-Theory Methods & Applications最新文献

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Fujita exponent and blow-up rate for a mixed local and nonlocal heat equation

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-02-03 DOI: 10.1016/j.na.2025.113761

Leandro M. Del Pezzo , Raúl Ferreira

In this paper we consider the blow-up problem for a mixed local-nonlocal diffusion operator,

u_{t} = a Δ u - b {(- Δ)}^{s} u + u^{p}

We show that the Fujita exponent is given by the nonlocal part,

p_{F} = 1 + 2 s / N

. We also determinate, in some cases, the blow-up rate.

引用次数: 0

Extremal functions for twisted sharp Sobolev inequalities with lower order remainder terms

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.na.2025.113758

Olivier Druet , Emmanuel Hebey

We prove existence of extremal functions and compactness of the set of extremal functions for twisted sharp 4-dimensional Sobolev inequalities with lower order remainder terms.

引用次数: 0

Universal differentiability sets in Laakso space

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-31 DOI: 10.1016/j.na.2025.113752

Sylvester Eriksson-Bique , Andrea Pinamonti , Gareth Speight

We show that there exists a family of mutually singular doubling measures on Laakso space with respect to which real-valued Lipschitz functions are almost everywhere differentiable. This implies that there exists a measure zero universal differentiability set in Laakso space. Additionally, we show that each of the measures constructed supports a Poincaré inequality.

引用次数: 0

Semiclassical states for the curl–curl problem

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-31 DOI: 10.1016/j.na.2025.113756

Bartosz Bieganowski , Adam Konysz , Jarosław Mederski

We show the existence of the so-called semiclassical states

U : R^{3} \to R^{3}

to the following curl–curl problem

ɛ^{2} \nabla \times (\nabla \times U) + V (x) U = g (U),

for sufficiently small

ɛ > 0

. We study the asymptotic behaviour of solutions as

ɛ \to 0^{+}

and we investigate also a related nonlinear Schrödinger equation involving a singular potential. The problem models large permeability nonlinear materials satisfying the system of Maxwell equations.

引用次数: 0

Radial positive solutions for mixed local and nonlocal supercritical Neumann problem

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-31 DOI: 10.1016/j.na.2025.113763

David Amundsen, Abbas Moameni, Remi Yvant Temgoua

In this paper, we establish the existence of positive non-decreasing radial solutions for a nonlinear mixed local and nonlocal Neumann problem in the ball. No growth assumption on the nonlinearity is required. We also provide a criterion for the existence of non-constant solutions provided the problem possesses a trivial constant solution.

引用次数: 0

Gradient Einstein-type warped products: Rigidity, existence and nonexistence results via a nonlinear PDE

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-30 DOI: 10.1016/j.na.2025.113759

José Nazareno Vieira Gomes , Willian Isao Tokura

We establish the necessary and sufficient conditions for constructing gradient Einstein-type warped metrics. One of these conditions leads us to a general Lichnerowicz equation with analytic and geometric coefficients for this class of metrics on the space of warping functions. In this way, we prove gradient estimates for positive solutions of a nonlinear elliptic differential equation on a complete Riemannian manifold with associated Bakry–Émery Ricci tensor bounded from below. As an application, we provide nonexistence and rigidity results for a large class of gradient Einstein-type warped metrics. Furthermore, we show how to construct gradient Einstein-type warped metrics, and then we give explicit examples which are not only meaningful in their own right, but also help to justify our results.

引用次数: 0

Asymptotics for positive singular solutions to subcritical sixth order equations

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-30 DOI: 10.1016/j.na.2025.113757

João Henrique Andrade , Juncheng Wei

We classify the local asymptotic behavior of positive singular solutions to a class of subcritical sixth order equations on the punctured ball. First, using a version of the integral moving spheres technique, we prove that solutions are asymptotically radially symmetric solutions with respect to the origin. We divide our approach into some cases concerning the growth of nonlinearity. In general, we use an Emden–Fowler change of variables to translate our problem to a cylinder. In the lower critical regime, this is not enough, thus, we need to introduce a new notion of change of variables. The main difficulty is that the cylindrical PDE in this coordinate system is nonautonomous. Nonetheless, we define an associated nonautonomous Pohozaev functional, which can be proved to be asymptotically monotone. In addition, we show a priori estimates for these two functionals, from which we extract compactness properties. With these ingredients, we can perform an asymptotic analysis technique to prove our main result.

引用次数: 0

Examples of optimal Hölder regularity in semilinear equations involving the fractional Laplacian

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-30 DOI: 10.1016/j.na.2025.113755

Gyula Csató , Albert Mas

We discuss the Hölder regularity of solutions to the semilinear equation involving the fractional Laplacian

{(- Δ)}^{s} u = f (u)

in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the nonlocality and the semilinearity of the equation, since it does not happen neither for local semilinear equations, nor for nonlocal linear equations. Namely, for nonlinearities

f

C^{β}

and when

2 s + β < 1

, the solution is not always

C^{2 s + β - ϵ}

for all

ϵ > 0

. Instead, in general the solution

u

is at most

C^{2 s / (1 - β)} .

引用次数: 0

Nonlinear stability of viscous shock profiles for a hyperbolic system with Cattaneo’s law and non-convex flux

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-24 DOI: 10.1016/j.na.2025.113749

Junyuan Deng , Lan Zhang

This paper is concerned with the large time behavior of solutions to the scalar conservation law with an artificial heat flux term. The heat flux is governed by Cattaneo’s law, which leads to a 2 × 2 system of hyperbolic equations. The existence and nonlinear stability of rarefaction waves and viscous shock waves have been derived under the assumption that flux function is strictly convex. In the current paper, we focus on the one-dimensional Cauchy problem for the system which allows for non-convex flux. Under Oleinik entropy condition, we obtain the existence and asymptotic stability of shifted viscous shock waves with sufficiently small wave strength. The proof is based on the standard energy method and shift theory.

引用次数: 0

Existence and boundedness of weak solutions to some vectorial Dirichlet problems

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-23 DOI: 10.1016/j.na.2025.113751

G.R. Cirmi , S. D’Asero , S. Leonardi , F. Leonetti , E. Rocha , V. Staicu

For integers

N \geq 2

n \geq 3

and

Ω

a bounded subset of

R^{n}

, we prove existence and boundedness of a weak solution

u

of the following prototype of nonlinear vectorial Dirichlet problem

(\begin{matrix} u \in W_{0}^{1, 2} (Ω, R^{N}) \\ - \sum_{i = 1}^{n} D_{i} A_{i}^{ν} (x, D u) = - \sum_{i = 1}^{n} D_{i} (E_{i}^{ν} (x) u^{ν}) + F^{ν} (x), \end{matrix})

for any

x \in Ω

and

ν = 1, 2, \dots, N

, where

u^{ν}

and

F^{ν}

denote the

ν -

th component of the vectors

u

and

F

, respectively, and the tensor

A (x, ξ)

satisfies suitable structural assumptions.

引用次数: 0

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