In this paper, we introduce the logarithmic fractional Sobolev space on Carnot groups and investigate the various functional-geometric aspects of the corresponding functional capacity. Moreover, we also investigate the logarithmic fractional perimeters and establish some fundamental properties of this perimeter, the coarea formula for logarithmic fractional Sobolev functions. Finally, we characterize the continuous embedding of the logarithmic fractional Sobolev space into the space L p(μ) in terms of the logarithmic fractional Sobolev capacity or the associated variational functional of a non-negative Radon measure μ or the logarithmic fractional perimeter.
Continuing our previous study (P. Drábek, S. Jung, E. Ko, and M. Zahradníková, Traveling waves for monostable reaction-diffusion-convection equations with discontinuous density-dependent coefficients, J. Math. Anal. Appl. 539 (2024), no. 1, part 1, 26) on the monostable reaction–diffusion–convection equation, we analyze the bistable case under weak regularity assumptions. Our approach applies monostable results on the subintervals where the reaction term g has constant sign, thereby establishing both existence and nonexistence of bistable traveling wave solutions. We extend the results of (L. Malaguti, C. Marcelli, and S. Matucci, Front propagation in bistable reaction-diffusion-advection equations, Adv. Differential Equations 9 (2004), no. 9–10, 1143–1166), obtained for p = 2 under higher regularity assumptions (d ∈ C 1[0, 1], g, h ∈ C[0, 1]), to the p-Laplacian with p > 1 in our weak regularity setting.
In this paper, we consider the uniformly fractional parabolic equations in a half space∂u∂t(x,t)+Dasu(x,t)=f(u(x,t)),(x,t)∈R+n×R, $$\frac{\partial u}{\partial t}\left(x,t\right)+{\mathcal{D}}_{a}^{s}u\left(x,t\right)=f\left(u\left(x,t\right)\right), \left(x,t\right)\in {\mathbb{R}}_{+}^{n}{\times}\mathbb{R},$$ for n ≥ 3. We mainly establish the monotonicity of solutions for the equations. Firstly, we derive the narrow region principle and maximum principle for antisymmetric functions under the assumptions that u is uniformly bounded and satisfies a growth condition, which weakens the conventional decay hypothesis u → 0 at infinity. In order to investigate the monotonicity of the solutions, we implement the method of moving planes.
Brendle [Sobolev inequalities in manifolds with nonnegative curvature, Comm. Pure Appl. Math. 76 (2022), no. 9, 2192–2218] successfully establishes the sharp Michael–Simon inequality for mean curvature on Riemannian manifolds with nonnegative sectional curvature (K≥0) $\left(\mathcal{K}\ge 0\right)$ via the Alexandrov–Bakelman–Pucci method. Nevertheless, this result cannot be extended to the hyperbolic space Hn+1 ${\mathbb{H}}^{n+1}$ (K=−1) $\left(\mathcal{K}=-1\right)$ , as a geodesic sphere provides a counterexample. In the present paper, we propose two conjectures concerning the hyperbolic version of the sharp Michael–Simon type inequality for k-th mean curvatures. However, the proof method posed by S. Brendle is not suitable for verifying the validity of these conjectures. This paper aims to utilize the curvature flow argument to prove the two conjectures for hypersurfaces with weaker convexity conditions. For k = 1, we first investigate a new locally constrained mean curvature flow in Hn+1 ${\mathbb{H}}^{n+1}$ and prove its longtime existence and exponential convergence. Then, the sharp Michael–Simon type inequality for the mean curvature of starshaped hypersurfaces in Hn+1 ${\mathbb{H}}^{n+1}$ is confirmed through the flow introduced in this paper. For k ≥ 2, the sharp Michael–Simon inequality for k-th mean curvatures of starshaped, strictly k-convex hypersurfaces in Hn+1 ${\mathbb{H}}^{n+1}$ is proven using the locally constrained inverse curvature flow introduced by Scheuer and Xia [Locally constrained inverse curvature flows, Trans. Amer. Math. Soc. 372 (2019), no. 10, 6771–6803].
The paper deals with the well-posedness of the time-dependent Stokes–Darcy model for an incompressible fluid in a bounded domain. The no-leak boundary condition and a multivalued nonmonotone slip condition of “friction type” is assumed. The latter is governed by the generalized subgradient of a locally Lipschitz superpotential. The transmission conditions across the interface boundary involve a new frictional boundary condition, which is proposed as a generalization of the popular Beavers–Joseph–Saffman law. The weak formulation of the problem is a parabolic hemivariational inequality for the Stokes velocity and the Darcy pressure. The well-posedness of the model is established under suitable assumptions involving a smallness condition.
This paper investigates a blow-up criterion for strong solutions to the Dirichlet problem of a compressible isentropic two-phase flow model under the influence of a magnetic field, allowing initial vacuum states. In this work, the BMO estimate for the Lamé system and a variant of the Brezis–Wainger’s inequality are introduced, which lead to a blow-up criterion expressed in terms of the Lt∞BMOx ${L}_{t}^{\infty }BM{O}_{x}$ -norm of the density and the Lt∞Lx307 ${L}_{t}^{\infty }{L}_{x}^{\frac{30}{7}}$ -norm of the magnetic field.
We consider a damped quasilinear equation of fourth order that models the mechanical vibrations of a marine riser. We study the nonexistence of global solutions, for any positive value of the initial energy. For this purpose we analyze a new differential inequality and construct a new positive invariant set, improving the results known in the literature. We analyze the influence of the damping term on the blow-up of the solution and find a finite critical damping coefficient for which the blow-up time becomes infinite. Finally, we propose a conjecture for the existence of global solutions for any positive value of the initial energy.
Using the wavelets characterization of inhomogeneous Lipschitz spaces, the author establishes a bilinear decomposition for products of functions in local Hardy spaces h p ( X ) ${h}<^>{p}\left(\mathcal{X} ight)$ and inhomogeneous Lipschitz spaces l i p 1 / p - 1 ( X ) ${\mathrm{l}\mathrm{i}\mathrm{p}}_{1/p-1}\left(\mathcal{X} ight)$ with p narrowly less than 1 on a space ( X , d , mu ) $\left(\mathcal{X},d,\mu ight)$ of homogeneous type, which is new even when X $\mathcal{X}$ is an Ahlfors regular space. Besides, the author obtains a sufficient condition and a necessary condition of the equivalence between the multipliers of l i p 1 / p - 1 ( X ) ${\mathrm{l}\mathrm{i}\mathrm{p}}_{1/p-1}\left(\mathcal{X} ight)$ and the intersection of the local bmo-type space b m o Phi p ( X ) ${\mathrm{b}\mathrm{m}\mathrm{o}}<^>{{{\Phi}}_{p}}\left(\mathcal{X} ight)$ and L infinity ( X ) ${L}<^>{\infty }\left(\mathcal{X} ight)$ . Using this, the author shows that this bilinear decomposition is sharp in the dual sense.
In this paper we prove some results on the compactness of solutions to the prescribed Webster scalar curvature problem on S3,θ0 $\left({\mathbb{S}}^{3},{\theta }_{0}\right)$ . We first combine blow-up analysis with the subcritical approximation method to precisely characterize the behavior of solutions at blow-up points. Then we further rule out the possibility of multiple blow-ups and establish the compactness results of the solutions. The crucial ingredient of our proofs is the characterize of blow up points, which is obtained by performing a blow up analysis procedure on the Heisenberg group.
We construct the L 2-gradient flow of the ideal functional, which is defined as the squared integral of the derivative of curvature, under the local length constraint. We prove that (i) the Cauchy problem of the gradient flow has a unique global-in-time solution, and (ii) the global-in-time solution converges to a critical point of the ideal functional under the total length constraint. As a corollary, we show the existence of a critical point with a rotation number of zero.
In this paper, we study an integrable Camassa–Holm (CH) type equation with quadratic nonlinearity. The CH type equation is shown integrable through a Lax pair, and particularly the equation is found to possess a new kind of peaked soliton (peakon) solution – called rogue peakon , that is given in a rational form with some logarithmic function, but not a regular traveling wave. We also provide multi-rogue peakon solutions. Furthermore, we discuss the local well-posedness of the solution in the Besov space B p , r s ${B}_{p,r}^{s}$ with 1 ≤ p , r ≤ ∞, s > max 1 + 1 / p , 3 / 2 $s{ >}\mathrm{max}\left\{1+1/p,3/2\right\}$ or B 2,1 3 / 2 ${B}_{2,1}^{3/2}$ , and then prove the ill-posedness of the solution in B 2 , ∞ 3 / 2 ${B}_{2,\infty }^{3/2}$ . Moreover, we establish the global existence and blow-up phenomenon of the solution, which is, if m 0 ( x ) = u 0 − u 0xx ≥ (≢)0, then the corresponding solution exists globally, meanwhile, if m 0 ( x ) ≤ (≢)0, then the corresponding solution blows up in a finite time.
In this paper, we consider the following nonlinear time-harmonic Maxwell system {del & times; del & times; E-1 - V-1(x)E-1 = mu(1)(x)|E-1|(p-2) E-1 + 2 lambda beta K(x)|E-2| (alpha)|E-1| (beta-2) E1, in & Ropf;(3) , del & times; del & times; E-2 - V-2(x)E-2 = mu(2)(x)|E-2|(p-2) E-2+ 2 lambda beta K(x)|E-2| (beta)|E-2| (beta-2) E2 , In & Ropf;(3, ) where del & times; denotes the curl operator, V-i , mu(i) and K : & Ropf;(3) -> & Ropf; are continuous functions with i = 1, 2. The constants satisfy lambda is an element of & Ropf;, 4 < p <= 6, alpha, beta > 2, and alpha + beta <= 6. For the subcritical growth case (i.e., 4 < p < 6, alpha + beta < 6), by using critical point theory and the Nehari-Pankov manifold, we first obtain a nontrivial ground state solution. And then, by exploiting the cylindrically symmetry of & Ropf;(3) and topological group methods, we establish the existence of a nontrivial cylindrically symmetric ground state solution. Finally, for the critical growth case (i.e., p = 6, alpha + beta = 6), we derive an unbounded sequence of nontrivial solutions.
In this work, we consider a flux-limited chemotaxis model with indirect signal production and nonlinear diffusion, {u(t) = del & sdot; (D(u)del u) - del & sdot;(uf(|del v|(2))del v) - k(1)u + k(2)w, x is an element of Omega, t > 0, 0 = Delta v - mu(t) + w, x is an element of Omega, t > 0, wt = Delta w - lambda(1)w + lambda(2)u, x is an element of Omega, t > 0 subjected to homogeneous Neumann boundary condition in a smoothly bounded domain Omega subset of & Ropf;(n) (n >= 3) , where k(1) >= 0, k(2) >= 0, lambda(1) > 0, lambda(2) > 0, mu(t) =1/|Omega| integral (Omega) wdx, f(zeta) = (1+zeta)(-alpha)(zeta >= 0, alpha > 0) , D(xi) similar or equal to xi(m-1) (m is an element of & Ropf;), xi similar or equal to infinity. If m > {2 - 4 / n - (1 - 3 / n)2 alpha for 0 < alpha <= 1/2, 1 - 1/n for alpha > 1/2, it is shown that the system admits a unique global classical solution, which is uniformly bounded when k(2)lambda(2) <= k(1)lambda(1). 0 < alpha <= 1/2, m < 2 - 4/n - (1 - 3/n)2 alpha, Moreover, if Omega = B-R(0) subset of & Ropf;(n) is a ball, and then there exist initial data such that the corresponding solution blows up at finite time. The above two results imply that for 0 < alpha <= 1/2, m < 2 - 4/n - (1 - 3/n)2 alpha determines a threshold. Especially, when n >= 5, alpha = n-4/2(n-3) serves as the critical blow-up threshold for D(u) = 1.
In this work, we study the two-dimensional anisotropic Boussinesq equations, incorporating only horizontal dissipation in the tangential velocity and horizontal thermal diffusion. When the spatial domain is T x R, this paper addresses the stability problem and characterizes the exact long-time dynamics of the perturbations without needing any structural assumptions on the initial data. Furthermore, we show that the vertical bar vertical bar(partial derivative(1)u, partial derivative(1)theta)vertical bar vertical bar(H1) and vertical bar vertical bar partial derivative(3)(1)theta vertical bar vertical bar(L2) decay exponentially in time. As a result, the H-1-norm of the oscillatory part ((u) over tilde, (theta) over tilde) also decays exponentially to zero, and thus (u, theta) converges to its horizontal average ((u) over bar, (theta) over bar). In particular, we improve the result of Dong-Wu-Xu-Zhu (Calc. Var. Partial Differ. Equations 60 (2021), no. 3, 116), where the horizontal dissipation for normal velocity is needed. This result illustrates the stratification phenomenon that is typical of buoyancy-driven fluids.
In this paper, we study the initial-boundary value problem for a viscoelastic Kirchhoff-like plate equation with logarithmic source term. Based on the contraction mapping principle, we prove the local existence and uniqueness of solutions. By the concavity arguments, we obtain the finite time blow-up and estimates on the upper bound for the blow-up time of solutions with negative, null, positive, and arbitrary positive initial energy, respectively.
In this paper, we construct and analyze a mosquito-borne disease model with chemotaxis in a homogeneous environment. Different from traditional modes that mosquitoes disperse with local diffusion, we correlate the chemotaxis of mosquitoes with the density of the infected human population and assume that mosquito diffusion is a decreasing function with respect to the density of infected human hosts. Our goal of this paper is to investigate how density-dependent diffusion affects the global dynamics of mosquito-borne diseases. We first prove the global existence and ultimate boundedness of the solution through a priori energy estimates. Then, we confirm that the basic reproduction number is a key threshold parameter that predicts whether the disease will persist or not. Moreover, we establish the global attractivity of the unique endemic steady state under some technical conditions. Our results demonstrate that the magnitude of the derivative of density-dependent diffusion function will remarkably affect the stability of the endemic steady state in the sense that when it is large enough, the endemic steady state will no longer be stable. This study also reveals that reducing the mosquito biting rate or controlling the movement of infected individuals can effectively disrupt the transmission chain, thereby providing a theoretical foundation for mosquito-borne disease prevention and control.
In this paper, we study multiple normalized solutions for the following Choquard equation: { -Delta u + V(x)u + lambda u = mu(I-alpha *|u|(p) ) u(p-2) u+ |u|(2*-2) u , in R-N integral(N)(R)-|u|(2) = alpha > 0 , where N >= 3, alpha is an element of (0, N), p is an element of ( 1 + (alpha) (N-2), 2* (alpha) , I-alpha is the Riesz potential, Vis an external potential vanishing at infinity and the parameter lambda arises as Lagrange multiplier. The purpose of this paper is to establish the existence of solutions with prescribed norm to the nonlinear equations in different assumptions. Under L-2-supercritical, L-2-critical perturbation mu(I-alpha * |u|(p) )u(p-2) u, respectively, we obtain the existence results and in L2-subcritical perturbation, we obtain multiple results, which one is a ground state normalized solution and others are normalized solutions with negative energy values. Furthermore, we consider the stability of the standing waves of the ground state and the tendency of u as alpha -> 0.
In this paper, we consider the chemotaxis-consumption system on a bounded smooth domain Omega subset of & Ropf;(n), n = 2, 3, with fluid coupling {p(t)+u center dot del p-del center dot(D(p)del p)=del center dot(pS(x,p,c)center dot del c) u.del c-Delta c=pc ut+k(u center dot del)u-Delta u+del pi=p del Phi, del center dot u=0. subject to the boundary conditions v center dot (D(p)del p + pS(x, p, c)del c)|partial derivative Omega = 0, (v center dot del c + c)partial derivative|Omega =gamma , and u|partial derivative Omega= 0. When (n, k) = (2, 1), we establish the global existence and uniform boundedness of classical solutions for all suitably regular initial data, under general structural conditions on the tensor-valued sensitivity S and a strictly positive lower bound on the diffusivity D. In case (n, k) = (3, 0), we show that the same result holds provided that D meets a certain diffusion enhancement condition depending on gamma. Moreover, we construct finite-time blow-up solutions for the radially symmetric, fluid-free system when n = 2, 3, D( ) less than or similar to (1 +xi )(m-1) with 0 < m < 2n and S equivalent to (n & times;n). We prove that, for any prescribed initial mass, blow-up occurs when gamma is sufficiently large.
In this paper, we study the strong instability of standing waves for the Hartree equation with a constant magnetic field. First, we prove the existence of least action ground states for the associated stationary equation using variational methods. Second, by establishing the approximation relation between the rescaled stationary magnetic Hartree equation and the classical Hartree equation, we prove the strong instability of least action ground state standing waves with positive angular momentum at sufficiently large frequencies in L 2-supercritical case.
