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Norm inflation for the viscous nonlinear wave equation Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-04-17 Pierre de Roubin, Mamoru Okamoto
In this article, we study the ill-posedness of the viscous nonlinear wave equation for any polynomial nonlinearity in negative Sobolev spaces. In particular, we prove a norm inflation result above the scaling critical regularity in some cases. We also show failure of \(C^k\)-continuity, for k the power of the nonlinearity, up to some regularity threshold.
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On the one-dimensional Pompeiu problem Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-04-15 Vivina Barutello, Camillo Costantini
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A global method for relaxation for multi-levelled structured deformations Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-04-13 Ana Cristina Barroso, José Matias, Elvira Zappale
We prove an integral representation result for a class of variational functionals appearing in the framework of hierarchical systems of structured deformations via a global method for relaxation. Some applications to specific relaxation problems are also provided.
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Fast rotation and inviscid limits for the SQG equation with general ill-prepared initial data Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-04-12 Gabriele Sbaiz, Leonardo Kosloff
In the present paper, we study the fast rotation and inviscid limits for the 2-D dissipative surface quasi-geostrophic equation with a dispersive forcing term, in the domain \(\Omega =\mathbb {T}^1\times \mathbb {R}\). In the case when we perform the fast rotation limit (keeping the viscosity fixed), in the context of general ill-prepared initial data, we prove that the limit dynamics is described
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Global solutions to the Kirchhoff equation with spectral gap data in the energy space Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-04-11 Marina Ghisi, Massimo Gobbino
We prove that the classical hyperbolic Kirchhoff equation admits global-in-time solutions for some classes of initial data in the energy space. We also show that there are enough such solutions so that every initial datum in the energy space is the sum of two initial data for which a global-in-time solution exists. The proof relies on the notion of spectral gap data, namely initial data whose components
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Global dynamics of a two-species clustering model with Lotka–Volterra competition Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-04-09 Weirun Tao, Zhi-An Wang, Wen Yang
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Existence of bound states for quasilinear elliptic problems involving critical growth and frequency Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-04-02
Abstract In this paper we study the existence of bound states for the following class of quasilinear problems, $$\begin{aligned} \left\{ \begin{aligned}&-\varepsilon ^p\Delta _pu+V(x)u^{p-1}=f(u)+u^{p^*-1},\ u>0,\ \text {in}\ {\mathbb {R}}^{N},\\&\lim _{|x|\rightarrow \infty }u(x) = 0, \end{aligned} \right. \end{aligned}$$ where \(\varepsilon >0\) is small, \(1
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Two-phase almost minimizers for a fractional free boundary problem Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-04-01 Mark Allen, Mariana Smit Vega Garcia
In this paper, we study almost minimizers to a fractional Alt–Caffarelli–Friedman type functional. Our main results concern the optimal \(C^{0,s}\) regularity of almost minimizers as well as the structure of the free boundary. We first prove that the two free boundaries \(F^+(u)=\partial \{u(\cdot ,0)>0\}\) and \(F^-(u)=\partial \{u(\cdot ,0)<0\}\) cannot touch, that is, \(F^+(u)\cap F^-(u)=\emptyset
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Semilinear elliptic problems in $$\mathbb {R}^N$$ : the interplay between the potential and the nonlinear term Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-03-28 Elves Alves de Barros e Silva, Sergio H. Monari Soares
It is considered a semilinear elliptic partial differential equation in \(\mathbb {R}^N\) with a potential that may vanish at infinity and a nonlinear term with subcritical growth. A positive solution is proved to exist depending on the interplay between the decay of the potential at infinity and the behavior of the nonlinear term at the origin. The proof is based on a penalization argument, variational
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Simple Lyapunov spectrum for linear homogeneous differential equations with $$L^p$$ parameters Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-03-24 Dinis Amaro, Mário Bessa, Helder Vilarinho
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A symmetry result for fully nonlinear problems in exterior domains Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-03-23 David Stolnicki
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Boundary value problems associated with Hamiltonian systems coupled with positively-(p, q)-homogeneous systems Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-03-20
Abstract We study the multiplicity of solutions for a two-point boundary value problem of Neumann type associated with a Hamiltonian system which couples a system with periodic Hamiltonian in the space variable with a second one with positively-(p, q)-homogeneous Hamiltonian. The periodic problem is also treated.
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Gradient higher integrability for singular parabolic double-phase systems Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-03-15
Abstract We prove a local higher integrability result for the gradient of a weak solution to parabolic double-phase systems of p-Laplace type when \(\tfrac{2n}{n+2}< p\le 2\) . The result is based on a reverse Hölder inequality in intrinsic cylinders combining p-intrinsic and (p, q)-intrinsic geometries. A singular scaling deficits affects the range of q.
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Singular solutions of semilinear elliptic equations with supercritical growth on Riemannian manifolds Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-03-14 Shoichi Hasegawa
In this paper, we shall discuss singular solutions of semilinear elliptic equations with general supercritical growth on spherically symmetric Riemannian manifolds. More precisely, we shall prove the existence, uniqueness and asymptotic behavior of the singular radial solution, and also show that regular radial solutions converges to the singular solution. In particular, we shall provide these properties
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Unified results for existence and compactness in the prescribed fractional Q-curvature problem Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-03-13 Yan Li, Zhongwei Tang, Heming Wang, Ning Zhou
In this paper we study the problem of prescribing fractional Q-curvature of order \(2\sigma \) for a conformal metric on the standard sphere \(\mathbb {S}^n\) with \(\sigma \in (0,n/2)\) and \(n\ge 3\). Compactness and existence results are obtained in terms of the flatness order \(\beta \) of the prescribed curvature function K. Making use of integral representations and perturbation result, we develop
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On backward Euler approximations for systems of conservation laws Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-03-12 Maria Teresa Chiri, Minyan Zhang
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A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-03-09
Abstract In this paper, we obtain the rate \(O(\varepsilon ^{1/2})\) of convergence in periodic homogenization of forced graphical mean curvature flows in the laminated setting. We also discuss with an example that a faster rate cannot be obtained by utilizing Lipscthiz estimates.
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Optimal control for the conformal CR sub-Laplacian obstacle problem Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-03-04 Pak Tung Ho, Cheikh Birahim Ndiaye
In this paper, we study an optimal control problem associated to the conformal CR sub-Laplacian obstacle problem on a compact pseudohermitian manifold. When the CR Yamabe constant is positive, we show that the optimal controls are equal to their associated optimal states and show the existence of a smooth optimal control which induces a conformal contact form with constant Webster scalar curvature
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Stable critical point of the Robin function and bubbling phenomenon for a slightly subcritical elliptic problem Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-20 Habib Fourti, Rabeh Ghoudi
In this paper, we deal with the boundary value problem \(-\Delta u= |u|^{4/(n-2)}u/[\ln (e+|u|)]^\varepsilon \) in a bounded smooth domain \( \Omega \) in \({\mathbb {R}}^n\), \(n\ge 3\) with homogenous Dirichlet boundary condition. Here \(\varepsilon >0\). Clapp et al. (J Differ Equ 275:418–446, 2021) built a family of solution blowing up if \(n\ge 4\) and \(\varepsilon \) small enough. They conjectured
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Three results on the energy conservation for the 3D Euler equations Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-20
Abstract We consider the 3D Euler equations for incompressible homogeneous fluids and we study the problem of energy conservation for weak solutions in the space-periodic case. First, we prove the energy conservation for a full scale of Besov spaces, by extending some classical results to a wider range of exponents. Next, we consider the energy conservation in the case of conditions on the gradient
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Multiplicity and concentration of solutions for a Choquard equation with critical exponential growth in $$\mathbb {R}^N$$ Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-19 Shengbing Deng, Xingliang Tian, Sihui Xiong
In this paper, we consider the following Choquard equation $$\begin{aligned} -\varepsilon ^{N}\Delta _{N}u+V(x)|u|^{N-2}u=\varepsilon ^{\mu -N}\left( I_\mu *F(u)\right) f(u) \quad {\text{ in }\quad \mathbb {R}^N}, \end{aligned}$$ where \(N\ge 3\), \(I_\mu =|x|^{-\mu }\) with \(0<\mu
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Multiplicity results for Hamiltonian systems with Neumann-type boundary conditions Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-15 Alessandro Fonda, Natnael Gezahegn Mamo, Franco Obersnel, Andrea Sfecci
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Nonlinear acoustic equations of fractional higher order at the singular limit Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-14 Vanja Nikolić
When high-frequency sound waves travel through media with anomalous diffusion, such as biological tissues, their motion can be described by nonlinear acoustic equations of fractional higher order. In this work, we relate them to the classical second-order acoustic equations and, in this sense, justify them as their approximations for small relaxation times. To this end, we perform a singular limit
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Existence and regularity results for nonlinear elliptic equations in Orlicz spaces Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-13 Giuseppina Barletta
We are concerned with the existence and regularity of the solutions to the Dirichlet problem, for a class of quasilinear elliptic equations driven by a general differential operator, depending on \((x,u,\nabla u)\), and with a convective term f. The assumptions on the members of the equation are formulated in terms of Young’s functions, therefore we work in the Orlicz-Sobolev spaces. After establishing
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Solutions of a quasilinear Schrödinger–Poisson system with linearly bounded nonlinearities Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-13
Abstract In this paper, we are concerned with the following quasilinear Schrödinger–Poisson system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+ K(x)\phi u=f(x,u),\quad &{}x\in {\mathbb {R}}^3,\\ -\Delta \phi -\varepsilon ^4\Delta _4\phi = K(x) u^2, &{}x\in {\mathbb {R}}^3, \end{array}\right. } \end{aligned}$$ where \(\varepsilon \) is a positive parameter and f is linearly bounded
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Embeddedness of min–max CMC hypersurfaces in manifolds with positive Ricci curvature Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-13 Costante Bellettini, Myles Workman
We prove that on a compact Riemannian manifold of dimension 3 or higher, with positive Ricci curvature, the Allen–Cahn min–max scheme in Bellettini and Wickramasekera (The Inhomogeneous Allen–Cahn Equation and the Existence of Prescribed-Mean-Curvature Hypersurfaces, 2020), with prescribing function taken to be a non-zero constant \(\lambda \), produces an embedded hypersurface of constant mean curvature
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Energy decay for wave equations with a potential and a localized damping Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-08 Xiaoyan Li, Ryo Ikehata
We consider the total energy decay together with the \(L^{2}\)-bound of the solution itself of the Cauchy problem for wave equations with a short-range potential and a localized damping, where we treat it in the one-dimensional Euclidean space \(\textbf{R}\). To study these, we adopt a simple multiplier method. In this case, it is essential that compactness of the support of the initial data not be
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Asymptotics for singular limits via phase functions Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-08
Abstract The asymptotic behavior of solutions as a small parameter tends to zero is determined for a variety of singular-limit PDEs. In some cases even existence for a time independent of the small parameter was not known previously. New examples for which uniform existence does not hold are also presented. Our methods include both an adaptation of geometric optics phase analysis to singular limits
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Partial regularity of minimizers for double phase functionals with variable exponents Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-02-01 Atsushi Tachikawa
The aim of this article is to study partial regularity of a minimizer \(\varvec{u:\Omega \subset \mathbb {R}^n \rightarrow \mathbb {R}^N}\) for a double phase functional with variable exponents: $$\begin{aligned} \varvec{\int \left( \vert Du\vert _A^{p(x)} + a(x) { \vert } Du{ \vert } _A^{q(x)}\right) dx,} \end{aligned}$$ where \(\varvec{{ \vert } \cdot { \vert }_A}\) stands for the norm deduced from
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A uniqueness criterion and a counterexample to regularity in an incompressible variational problem Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-01-27 M. Dengler, J. J. Bevan
In this paper we consider the problem of minimizing functionals of the form \(E(u)=\int _B f(x,\nabla u) \,dx\) in a suitably prepared class of incompressible, planar maps \(u: B \rightarrow \mathbb {R}^2\). Here, B is the unit disk and \(f(x,\xi )\) is quadratic and convex in \(\xi \). It is shown that if u is a stationary point of E in a sense that is made clear in the paper, then u is a unique global
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The move from Fujita type exponent to a shift of it for a class of semilinear evolution equations with time-dependent damping Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-01-27
Abstract In this paper, we derive suitable optimal \(L^p-L^q\) decay estimates, \(1\le p\le 2\le q\le \infty \) , for the solutions to the \(\sigma \) -evolution equation, \(\sigma >1\) , with scale-invariant time-dependent damping and power nonlinearity \(|u|^p\) , $$\begin{aligned} u_{tt}+(-\Delta )^\sigma u + \frac{\mu }{1+t} u_t= |u|^{p}, \quad t\ge 0, \quad x\in {{\mathbb {R}}}^n, \end{aligned}$$
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Annular type surfaces with fixed boundary and with prescribed, almost constant mean curvature Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-01-25 Paolo Caldiroli, Gabriele Cora, Alessandro Iacopetti
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On radial positive normalized solutions of the Nonlinear Schrödinger equation in an annulus Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-01-22 Jian Liang, Linjie Song
We are interested in the following semilinear elliptic problem: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + \lambda u = u^{p-1}, x \in T, \\ u > 0, u = 0 \ \text {on} \ \partial T, \\ \int _{T}u^{2} \, dx= c \end{array}\right. } \end{aligned}$$ where \(T = \{x \in \mathbb {R}^{N}: 1< |x| < 2\}\) is an annulus in \(\mathbb {R}^{N}\), \(N \ge 2\), \(p > 1\) is Sobolev-subcritical, searching
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Fujita-type results for the degenerate parabolic equations on the Heisenberg groups Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-01-20 Ahmad Z. Fino, Michael Ruzhansky, Berikbol T. Torebek
In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted
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Optimization of the Dirichlet problem for gradient differential inclusions Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-01-20 Elimhan N. Mahmudov, Dilara Mastaliyeva
The paper is devoted to optimization of the gradient differential inclusions (DFIs) on a rectangular area. The discretization method is the main method for solving the proposed boundary value problem. For the transition from discrete to continuous, a specially proven equivalence theorem is provided. To optimize the posed continuous gradient DFIs, a passage to the limit is required in the discrete-approximate
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On a characterization of the Rellich–Kondrachov theorem on groups and the Bloch spectral cell equation Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-01-03 Vernny Ccajma, Wladimir Neves, Jean Silva
This paper is concerned with the Rellich–Kondrachov Theorem on Groups. We establish some conditions which characterize in a precise manner important properties related to this theorem and the Sobolev spaces on groups involved on it. The main motivation to study the Rellich–Kondrachov Theorem on Groups comes from the Bloch spectral cell equation, which is an eigenvalue-eigenfunction problem associated
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Boundedness of solutions to a chemotaxis–haptotaxis model with nonlocal terms Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2024-01-06 Guoqiang Ren
In this paper, we consider the chemotaxis–haptotaxis model of two different types (parabolic–elliptic, fully parabolic) with nonlocal terms under Neumann boundary conditions in a bounded domain with smooth boundary. We show that the system possesses a unique global classical solution in different cases. Our results generalize and improve partial previously known ones.
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Liouville-type theorems for fractional Hardy–Hénon systems Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-12-28 Kui Li, Meng Yu, Zhitao Zhang
In this paper, we study Liouville-type theorems for fractional Hardy–Hénon elliptic systems with weights. Because the weights are singular at zero, we firstly prove that classical solutions for systems in \({\mathbb {R}}^N \backslash \{0\}\) are also distributional solutions in \({\mathbb {R}}^N\). Then we study the equivalence between the fractional Hardy–Hénon system and a proper integral system
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The continuous dependence of the viscous Boussinesq equations uniformly with respect to the viscosity Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-12-27 Rong Chen, Zhichun Yang, Shouming Zhou
Abstract This paper focuses on the inviscid limit of the incompressible Boussinesq equations in the same topology as the initial data, and proved that the continuous dependence of the viscous Boussinesq equations uniformly in some Besov spaces with respect to the viscosity. Our results extends the work of Guo et al. (J Funct Anal 276(9):2821–2830, 2019) on Navier–Stokes equations to Boussinesq equations
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Evolutionary stable strategies and cubic vector fields Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-12-26 Jefferson Bastos, Claudio Buzzi, Paulo Santana
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Existence and non-existence results for cooperative elliptic systems without variational structure Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-12-15 John Villavert
We consider general cooperative elliptic systems possibly without variational structure and with differential operator resembling that from an Euler–Lagrange equation for a sharp Hardy–Sobolev inequality. Under suitable growth conditions on the source nonlinearities and geometric assumptions on the domain, we derive various existence and non-existence results and Liouville theorems. The results are
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Well-posedness of Whitham-Broer-Kaup equation with negative dispersion Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-12-15 Nabil Bedjaoui, Youcef Mammeri
In this work, we discuss the well-posedness of Whitham-Broer-Kaup equation with negative dispersion term. A symmetrizer is built, then we prove the existence and uniqueness of a solution using the vanishing viscosity method.
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A singular perturbation problem for a nonlinear Schrödinger system with three wave interaction Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-12-12 Yuki Osada
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q-Laplace equation involving the gradient on general bounded and exterior domains Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-12-12 A. Razani, C. Cowan
The existence of positive singular solutions of $$\begin{aligned} \left\{ \begin{array}{lcc} -\Delta _q u=(1+g(x))|\nabla u|^p &{}\quad \text {in}&{} B_1,\\ u=0&{}\quad \text {on}&{} \partial B_1, \end{array} \right. \end{aligned}$$(1) is proved, where \(B_1\) is the unit ball in \({\mathbb {R}}^N\), \(N \ge 3\), \(2 0\) for large |x| is proved, in the case of \(\Omega \) an exterior domain \({\mathbb
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The Cheeger cut and Cheeger problem in metric measure spaces Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-12-13 José M. Mazón
In this paper we study the Cheeger cut and Cheeger problem in the general framework of metric measure spaces. A central motivation for developing our results has been the desire to unify the assumptions and methods employed in various specific spaces, such as Riemannian manifolds, Heisenberg groups, graphs, etc. We obtain two characterization of the Cheeger constant: a variational one and another one
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Global well-posedness for eddy-mean vorticity equations on $$\mathbb {T}^2$$ Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-12-12 Yuri Cacchio’
We consider the two-dimensional, \(\beta \)-plane, eddy-mean vorticity equations for an incompressible flow, where the zonally averaged flow varies on scales much larger than the perturbation. We prove global existence and uniqueness of the solution to the equations on periodic settings.
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On weak (measure valued)–strong uniqueness for Navier–Stokes–Fourier system with Dirichlet boundary condition Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-12-11 Nilasis Chaudhuri
In this article, our goal is to define a measure valued solution of compressible Navier–Stokes–Fourier system for a heat conducting fluid with Dirichlet boundary condition for temperature in a bounded domain. The definition will be based on the weak formulation of entropy inequality and ballistic energy inequality. Moreover, we obtain the weak (measure valued)–strong uniqueness property of this solution
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A damped elastodynamics system under the global injectivity condition: local wellposedness in $$L^p$$ -spaces Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-11-04 Sébastien Court
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Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-10-26 Davide Barilari, Karen Habermann
We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace–Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field
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A family of nonlocal degenerate operators: maximum principles and related properties Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-10-19 Delia Schiera
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Pushed fronts in a Fisher–KPP–Burgers system using geometric desingularization Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-10-20 Matt Holzer, Matthew Kearney, Samuel Molseed, Katie Tuttle, David Wigginton
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A Relation of the Allen–Cahn equations and the Euler equations and applications of the equipartition Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-10-06 Dimitrios Gazoulis
We will prove that solutions of the Allen–Cahn equations that satisfy the equipartition of the energy can be transformed into solutions of the Euler equations with constant pressure. As a consequence, we obtain De Giorgi type results, that is, the level sets of entire solutions are hyperplanes. Also, we will determine the structure of solutions of the Allen–Cahn system in two dimensions that satisfy
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Existence and non-existence results for a semilinear fractional Neumann problem Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-10-03 Eleonora Cinti, Francesca Colasuonno
We establish a priori \(L^\infty \)-estimates for non-negative solutions of a semilinear nonlocal Neumann problem. As a consequence of these estimates, we get non-existence of non-constant solutions under suitable assumptions on the diffusion coefficient and on the nonlinearity. Moreover, we prove an existence result for radial, radially non-decreasing solutions in the case of a possible supercritical
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Harmonic embeddings of the Stretched Sierpinski Gasket Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-10-04 Ugo Bessi
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Nonlinear dynamic problems for 2D magnetoelastic waves Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-09-25 Viatcheslav Priimenko, Mikhail Vishnevskii
The propagation of magnetoelastic waves in a two-dimensional electroconductive elastic body is investigated. The waves are fully coupled through the nonlinear magnetoelastic effect. We prove the existence and uniqueness for both the forward problem and the inverse problem, which consists of identifying the unknown scalar time-dependent component in the body density force acting on the elastic body
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Asymptotic mean value properties for the elliptic and parabolic double phase equations Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-09-20 Weili Meng, Chao Zhang
We characterize an asymptotic mean value formula in the viscosity sense for the double phase elliptic equation $$\begin{aligned} -{\textrm{div}}(|\nabla u |^{p-2}\nabla u+ a(x)|\nabla u |^{q-2}\nabla u)=0 \end{aligned}$$ and the normalized double phase parabolic equation $$\begin{aligned} u_t=|\nabla u |^{2-p}{\textrm{div}}(|\nabla u |^{p-2}\nabla u+ a(x,t)|\nabla u |^{q-2}\nabla u), \quad 1
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The bounded slope condition for parabolic equations with time-dependent integrands Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-09-12 Leah Schätzler, Jarkko Siltakoski
In this paper, we study the Cauchy–Dirichlet problem $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {\text {div}} \left( D_\xi f(t, Du)\right) = 0 &{} \quad \hbox {in} \ \Omega _T, \\ u = u_o &{} \quad \hbox { on} \ \partial _{\mathcal {P}} \Omega _T,\\ \end{array} \right. \end{aligned}$$ where \(\Omega \subset \mathbb {R}^n\) is a convex and bounded domain, \(f:[0,T]\times {\mathbb {R}}^n
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The Brezis–Nirenberg problem for systems involving divergence-form operators Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-09-08 Burton Brown, Mathew Gluck, Vince Guingona, Thomas Hammons, Miriam Parnes, Sean Pooley, Avery Schweitzer
We study a system of nonlinear elliptic partial differential equations involving divergence-form operators. The problem under consideration is a natural generalization of the classical Brezis–Nirenberg problem. We find conditions on the domain, the coupling coefficients and the coefficients of the differential operator under which positive solutions are guaranteed to exist and conditions on these objects
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Velocity diagram of traveling waves for discrete reaction–diffusion equations Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-08-31 M. Al Haj, R. Monneau
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Feynman–Kac formula for BSDEs with jumps and time delayed generators associated to path-dependent nonlinear Kolmogorov equations Nonlinear Differ. Equ. Appl. (IF 1.2) Pub Date : 2023-08-29 Luca Di Persio, Matteo Garbelli, Adrian Zălinescu
We consider a system of forward backward stochastic differential equations (FBSDEs) with a time-delayed generator driven by Lévy-type noise. We establish a non-linear Feynman–Kac representation formula associating the solution given by the FBSDEs system to the solution of a path dependent nonlinear Kolmogorov equation with both delay and jumps. Obtained results are then applied to study a generalization