In this work, we investigate a two-species chemotaxis system with singular sensitivity and indirect signal production in an n-dimensional domain (n≥ 1) . The movement of each species is governed by a nonlinear sensitivity function that responds to chemical gradients. The chemical substances interact dynamically: one chemical is secreted by the species, while the other degrades over time and influences the movement. The model also incorporates logistic growth and interspecific competition between the species. Using semigroup theory and a Moser-type iteration framework, we establish the global boundedness of classical solutions.
In this paper, we first derive area estimates for weighted stable minimal capillary surfaces in a 3-dimensional weighted Riemannian manifold M with boundary ∂M , in terms of the weighted scalar curvature of M and the weighted mean curvature of ∂M . As an application of these estimates, we establish a rigidity theorem for 3-dimensional weighted Riemannian manifolds in the equality case. Finally, we investigate the relationship between the weighted scalar curvature of M (or the weighted mean curvature of ∂M ) and the topological properties of weighted stable minimal capillary surfaces.
Abstract In this paper we present a new global $${L^\infty }$$ L ∞ -estimate for solutions $$u\in D^{s,p}({\mathbb R}^N)$$ u ∈ D s , p ( R N ) of the fractional p -Laplacian equation $$ u\in D^{s,p}({\mathbb R}^N): (-\Delta _p)^s u=f(x,u) \quad \text{ in } {\mathbb R}^N, $$ u ∈ D s , p ( R N ) : ( - Δ p ) s u = f ( x , u ) in R N , of the form $$ \Vert u\Vert _{\infty }\le C \Phi (\Vert u\Vert _{\beta }) $$ ‖ u ‖ ∞ ≤ C Φ ( ‖ u ‖ β ) for some $$\beta > p$$ β > p , where $$\Phi : {\mathbb R}^+\rightarrow {\mathbb R}^+$$ Φ : R + → R + is a data independent function with $$\lim _{s\rightarrow 0^+}\Phi (s)=0$$ lim s → 0 + Φ ( s ) = 0 . The obtained $$L^\infty $$ L ∞ -estimate is used to prove a decay estimate based on pointwise estimates in terms of nonlinear Wolff potentials. Taking advantage of both the $$L^\infty $$ L ∞ and decay estimate we prove a Brezis-Nirenberg type result regarding $$D^{s,2}({\mathbb R}^N)$$ D s , 2 ( R N ) versus $$C_b\left( {\mathbb R}^N, 1+|x|^{N-2s}\right) $$ C b R N , 1 + | x | N - 2 s local minimizers.
We characterize the well-posedness of the higher order regularity problem in the upper half-space with data in Sobolev Banach function spaces by proving its equivalence to natural weighted estimates for the Hardy–Littlewood maximal operator. The generality of our framework allows for applications to Lebesgue spaces, rearrangement-invariant spaces such as Orlicz spaces, and variable exponent Lebesgue spaces, as well as their weighted counterparts, among others. This is established for the family of second-order, homogeneous, elliptic, constant complex coefficient systems in ℝ^n that admit a distinguished coefficient tensor, a natural condition that always holds in the scalar case and for the Lamé system of elasticity.
We give upper bounds for the Poincaré and logarithmic Sobolev constants for doubly weighted Brownian motion on manifolds with sticky-reflecting boundary diffusion under curvature assumptions on the manifold and its boundary. To achieve this we use an interpolation approach based on energy interactions between the boundary and the interior of the manifold as well as the weighted Reilly formula. Along the way we also obtain a lower bound on the first nontrivial doubly weighted Steklov eigenvalue and an upper bound on the norm of the doubly weighted boundary trace operator on Sobolev functions. We also consider the case of doubly weighted Brownian motion with pure sticky reflection.
In this paper, we investigate the strong convergence of Wong-Zakai approximations for stochastic differential equations (SDEs) with oblique reflection in non-smooth time-dependent domains, as introduced in [Stochastic Process. Appl. 121 (2011), no. 7, 1464–1491]. Specifically, we establish that the sequence of adapted approximate solutions to the Wong-Zakai equations converges in the L^p -sense uniformly on [0, T] to the unique solution of the reflected SDE in such domains.
In this paper we analyze the asymptotic behaviour as p→ 1^+ of solutions u_p to {[ -Δ _pu_p = λ |∇ u_p|^p-2∇ u_p·x/|x|^2+ f in Ω ,; u_p = 0 on ∂Ω , ]. where Ω is a bounded open subset of ℝ^N with Lipschitz boundary containing the origin, λ∈ℝ , and f is a nonnegative datum in L^N,∞(Ω ) . As a consequence, under suitable smallness assumptions on f and λ , we show sharp existence results of bounded solutions to the Dirichlet problems {[ - Δ _1 u = λD u/|D u|·x/|x|^2+f in Ω ,; u=0 on ∂Ω , ]. where Δ _1u=div ( Du/|Du|) is the 1-Laplacian operator. The case of a generic drift term in L^N,∞(Ω ) is also considered. Explicits examples are given in order to show the optimality of the main assumptions on the data.
We study the nonlinear discrete random Schrödinger equation i ∂/∂ t u+(ε _n Δ +V) u+δ |u|^2 p u =0 ( p ∈ℕ^+) and discrete random wave equation u_tt+(ε _n Δ +V) u+δ u^2 p+1 =0 (p∈ℕ^+) on ℤ^d× [0, ∞ ) , where 0< δ <ε≪ 1, Δ is the discrete Laplacian and V is the random potential, | ε _n| ≤ε e^-ϱ |n| with ϱ >0 . We fix the random potential V in a good set and choose the small amplitudes as parameters to prove a KAM theorem for finding linearly stable quasi-periodic solutions of nonlinear random Schrödinger equation and wave equation.
Consider the following Hénon type system: 0.1 {[ -Δ u = |x|^α v^p, in B_1(0),; -Δ v = |x|^α u^q, in B_1(0),; u = v= 0, on ∂ B_1(0), ]. where B_1(0)⊂ℝ^N is the unit ball centered at the origin, N≥ 3 , α > 0 and p, q>1 satisfy: 1p+1 + 1q+1 > N-2N. It is shown that the ground state solution of Eq. 0.1 exists for each α >0 and satisfies the foliated Schwartz symmetry property when α is large. In this paper, we first investigate the asymptotic behavior of the ground state solution ( u_α , v_α ) for Eq. 0.1, as α→ +∞ . Then we analyse the profile of the solutions with one peak or multiple peaks.
In this paper, we investigate the bilinear Stein’s square functions associated with the bilinear Bochner-Riesz means, defined as 𝒢^α (f, g)(x):=( ∫ _0^∞| ∂/∂ Rℬ_R^α +1(f, g)(x)| ^2 R d R) ^1/2, where the bilinear Bochner-Riesz means is given by ℬ_R^α (f, g)(x)=∫ _ℝ^n∫ _ℝ^n( 1-|ξ |^2+|η |^2/R^2) _+^αf̂(ξ ) ĝ(η ) e^2 π i x · (ξ +η ) d ξ d η , R>0. The weighted strong and weak type estimates for the operator 𝒢^α and its associated commutators are given. We also show that the commutators [b⃗,𝒢^α ] are compact from L^p_1(w_1)× L^p_2(w_2) to L^p(v_w⃗) for 1n-1/2 , where b⃗=(b_1,b_2) , b_1,b_2∈ CMO (ℝ^n) , w⃗=( w_1, w_2) ∈A_p⃗ and v_w⃗=w_1^p/p_1 w_2^p/ p_2. Here CMO(ℝ^n) is the closure of C_c^∞ (ℝ^n) in the BMO(ℝ^n) topology.
Let n≥ 2 , Ω⊂ℝ^n be a bounded non-tangentially accessible domain (for short, NTA domain), and p(· ):ℝ^n→ (0,∞ ) a variable exponent function satisfying 0
We consider the Grushin operator with drift which is symmetric with respect to a measure having exponential growth. For the corresponding Riesz transforms, we study strong-type (p, p), 1< p < ∞ , and weak-type (1, 1) boundedness.
In the foundational contributions of Ricci and Stein (Ann. Inst. Fourier (Grenoble) 42, 637–670, 1992) and Fefferman and Pipher (Amer. J. Math. 119, 337–369, 1997), the authors develeped the L^p, 1
In this paper we study the balayage for weighted Bergman spaces on the unit disk induced by doubling radial weights. The focus is on the relationship between properties of the balayage and its inducing measure. We also study the relationship between the balayage and the Berezin transform of a positive Borel measure.
We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a planar, simply connected domain. The eigenvalues admit a characterization in terms of a layer potential of the Helmholtz equation. Using the exterior conformal mapping associated with the given domain, we reformulate the layer potential as an infinite-dimensional matrix. Based on this matrix representation, we develop a finite section approach for approximating the Laplacian eigenvalues and provide a convergence analysis by applying the Gohberg–Sigal theory for operator-valued functions. Moreover, we derive an asymptotic formula for the Laplacian eigenvalues on deformed domains that results from the changes in the conformal mapping coefficients.
In this paper, we consider the averaging principle for stochastic differential equations (SDEs) with irregular coefficients. The structure of the article is roughly divided into two parts. Firstly, we use the regularity of the non-degenerate Kolmogorov equation to establish the averaging principle for a class of SDEs with drift coefficients satisfying the Dini continuity condition. Secondly, when the diffusion coefficient is only Hölder continuous in space variable, we obtain corresponding results using the Yamada–Watanabe approximation techniques.
We consider the Dirichlet problem for a class of linear elliptic systems with their coefficients being singular or degenerate in one-dimensional variable over a bounded rough domain. Under assumptions that the coefficient meets the partial small weighted BMO semi-norm and the boundary of the underlying domain is a bounded Reifenberg flatness, we prove global estimates of the gradients in the weighted generalized Morrey spaces to this problem including the borderline setting.
