Volume 24, Issue 4

In this paper, we are devoted to establishing the point-wise decay estimates for solution to the 5D defocusing energy-critical Hartree equation with an initial data in H 2 ( R 5 ) ∩ L 1 ( R 5 ) ${H}^{2}({\mathbb{R}}^{5})\cap {L}^{1}({\mathbb{R}}^{5})$ . We show that the nonlinear solution has the same time decay rate as the linear one. The main new ingredient is that we used the theories of Lorentz spaces to overcome the low power of nonlinearity.

We consider nonlocal differential equations with convolution coefficients of the form − M ( a * | u | q ) ( 1 ) μ ( t ) u ″ ( t ) = λ f t , u ( t ) ,  t ∈ ( 0,1 ) , $$-M\left(\left(a {\ast} \vert u{\vert }^{q}\right)\left(1\right)\mu \left(t\right)\right){u}^{{\prime\prime}}\left(t\right)=\lambda f\left(t,u\left(t\right)\right)\text{,\,}t\in \left(0,1\right),$$ where q > 0, subject to given boundary data. The function μ ∈ C [ 0,1 ] $\mu \in \mathcal{C}\left(\left[0,1\right]\right)$ modulates the strength of the nonlocal element. We demonstrate that the nonlocality has a strong deregularising effect in the specific sense that nonexistence theorems for this problem are directly affected by the magnitude of the function μ . A specific example illustrates the application of the nonexistence results presented herein.

Resorting to the spectral analysis of the 4 × 4 matrix spectral problem, we construct a 4 × 4 matrix Riemann–Hilbert problem to solve the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation. The nonlinear steepest decent method is extended to study the 4 × 4 matrix Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems, the basic Riemann–Hilbert problem is reduced to a model Riemann–Hilbert problem, by which the long-time asymptotic behavior to the solution of the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation is obtained with the help of the asymptotic expansion of the parabolic cylinder function and strict error estimates.

We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation for the displacement and a parabolic equation for the temperature of the elastic material. We allow the hyperbolic equation to degenerate at spacial infinity, which results that the coefficients of the coupled system are not uniformly bounded and then the previous methods for the strictly hyperbolic-parabolic coupled systems are invalid. We introduce a suitable weighted norm to establish the local existence and uniqueness of classical solutions by the contraction mapping principle. The existence time T of the solution is independent of the spatial variable.

For a probability P in R d ${\mathbb{R}}^{d}$ its center outward distribution function F ± , introduced in V. Chernozhukov, A. Galichon, M. Hallin, and M. Henry (“Monge–Kantorovich depth, quantiles, ranks and signs,” Ann. Stat. , vol. 45, no. 1, pp. 223–256, 2017) and M. Hallin, E. del Barrio, J. Cuesta-Albertos, and C. Matrán (“Distribution and quantile functions, ranks and signs in dimension d: a measure transportation approach,” Ann. Stat. , vol. 49, no. 2, pp. 1139–1165, 2021), is a new and successful concept of multivariate distribution function based on mass transportation theory. This work proves, for a probability P with density locally bounded away from zero and infinity in its support, the continuity of the center-outward map on the interior of the support of P and the continuity of its inverse, the quantile, Q ± . This relaxes the convexity assumption in E. del Barrio, A. González-Sanz, and M. Hallin (“A note on the regularity of optimal-transport-based center-outward distribution and quantile functions,” J. Multivariate Anal. , vol. 180, p. 104671, 2020). Some important consequences of this continuity are Glivenko–Cantelli type theorems and characterisation of weak convergence by the stability of the center-outward map.

We investigate the following fractional p -Laplacian convex-concave problem: ( P λ ) ( − Δ ) p s u = λ | u | q − 2 u + | u | p s * − 2 u   in Ω , u = 0   in R n \ Ω ,   $$\left({P}_{\lambda }\right) \begin{cases}\begin{aligned}\hfill {\left(-{\Delta}\right)}_{p}^{s}u& =\lambda \vert u{\vert }^{q-2}u+\vert u{\vert }^{{p}_{s}^{{\ast}}-2}u\hfill & \hfill & \quad \text{in} {\Omega},\hfill \\ \hfill u& =0 \hfill & \hfill & \quad \text{in} {\mathbb{R}}^{n}{\backslash}{\Omega},\hfill \end{aligned}\quad \hfill \end{cases}$$ where Ω is a bounded C 1,1 domain in R n ${\mathbb{R}}^{n}$ , s ∈ (0, 1), p > q > 1, n > sp , λ > 0, and p s * = n p n − s p ${p}_{s}^{{\ast}}=\frac{np}{n-sp}$ is the critical Sobolev exponent. Our analysis extends classical works (A. Ambrosetti, H. Brezis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,” J. Funct. Anal. , vol. 122, no. 2, pp. 519–543, 1994, B. Barrios, E. Colorado, R. Servadei, and F. Soria, “A critical fractional equation with concave-convex power nonlinearities,” Ann. Inst. Henri Poincare Anal. Non Lineaire , vol. 32, no. 4, pp. 875–900, 2015, J. García Azorero, J. Manfredi, and I. Peral Alonso, “Sobolev versus Hölder local minimizer and global multiplicity for some quasilinear elliptic equations,” Commun. Contemp. Math. , vol. 2, no. 3, pp. 385–404, 2000) to fractional p -Laplacian. Owing to the nonlinear and nonlocal properties of ( − Δ ) p s ${\left(-{\Delta}\right)}_{p}^{s}$ , we need to overcome many difficulties and apply notably different approaches, due to the lack of Picone identity, the stability theory, and the strong comparison principle. We show first a dichotomy result: a positive W 0 s , p ( Ω ) ${W}_{0}^{s,p}\left({\Omega}\right)$ solution of ( P λ ) exists if and only if λ ∈ (0, Λ] with an extremal value Λ ∈ (0, ∞). The W 0 s , p ( Ω ) ${W}_{0}^{s,p}\left({\Omega}\right)$ regularity for the extremal solution seems to be unknown regardless of whether s = 1 or s ∈ (0, 1). When p ≥ 2, p − 1 < q < p and n > s p ( q + 1 ) q + 1 − p $n{ >}\frac{sp\left(q+1\right)}{q+1-p}$ , we get two positive solutions for ( P λ ) with small λ > 0. Here the mountain pass structure is more involved than the classical situations due to the lack of explicit minimizers for the Sobolev embedding, we should proceed carefully and simultaneously the construction of mountain pass geometry and the estimate for mountain pass level. Finally, we show another new result for ( P λ ) and all p > q > 1: without sign constraint, ( P λ ) possesses infinitely many solutions when λ > 0 is small enough. Here we use the Z 2 ${\mathbb{Z}}_{2}$ -genus theory, based on a space decomposition for reflexible and separable Banach spaces, which has its own interest.

This paper is concerned with the periodic solutions for a coupled system of wave equations with x -dependent coefficients. Such a model arises naturally when two waves propagate simultaneously in the nonisotrpic media. In this paper, for the periods having the form T = 2 a − 1 b ( a , b are positive integers ) $T=\frac{2a-1}{b}\left(a,b \text{are\,positive\,integers}\right)$ and some types of boundary conditions, we obtain the existence of the time periodic solutions and analyze the asymptotic behaviors as the coupled parameter goes to zero, when the nonlinearities are superlinear and monotone, by using the variational method. In particular, the condition ess inf  η ϱ ( x ) > 0 is not required.

In this paper, we consider the free boundary problem of the radially symmetric compressible Navier–Stokes equations with viscosity coefficients of the form μ ( ρ ) = ρ θ , λ ( ρ ) = ( θ − 1) ρ θ in R N ${\mathbb{R}}^{N}$ . Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, “Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries,” J. Differ. Equ. , vol. 253, no. 1, pp. 1–19, 2012) for N = 3, θ = γ > 1 and improve the spreading rate of the free boundary, where γ is the adiabatic exponent. Moreover, we construct an analytical solution for θ = 2 3 $\theta =\frac{2}{3}$ , N = 3 and γ > 1, and we prove that the free boundary grows linearly in time by using some new techniques. When θ = 1, under the stress free boundary condition, we construct some analytical solutions for N = 2, γ = 2 and N = 3, γ = 5 3 $\gamma =\frac{5}{3}$ , respectively.

In this work, we prove a two-scale homogenization result for a set of diffusion-coagulation Smoluchowski-type equations with transmission boundary conditions. This system is meant to describe the aggregation and diffusion of pathological tau proteins in the cerebral tissue, a process associated with the onset and evolution of a large variety of tauopathies (such as Alzheimer’s disease). We prove the existence, uniqueness, positivity and boundedness of solutions to the model equations derived at the microscale (that is the scale of single neurons). Then, we study the convergence of the homogenization process to the solution of a macro-model asymptotically consistent with the microscopic one.

In this paper, we study the following fractional Schrödinger–Poisson system with discontinuous nonlinearity: ε 2 s ( − Δ ) s u + V ( x ) u + ϕ u = H ( u − β ) f ( u ) , in R 3 , ε 2 s ( − Δ ) s ϕ = u 2 , in R 3 , u > 0 , in R 3 , $$\begin{cases}^{2s}{\left(-{\Delta}\right)}^{s}u+V\left(x\right)u+\phi u=H\left(u-\beta \right)f\left(u\right),\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ {\varepsilon }^{2s}{\left(-{\Delta}\right)}^{s}\phi ={u}^{2},\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ u{ >}0,\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \end{cases}$$ where ɛ > 0 is a small parameter, s ∈ ( 3 4 , 1 ) $s\in \left(\frac{3}{4},1\right)$ , β > 0, H is the Heaviside function, (−Δ) s u is the fractional Laplacian operator, V : R 3 → R $V :{\mathbb{R}}^{3}\to \mathbb{R}$ is a continuous potential and f : R → R $f :\mathbb{R}\to \mathbb{R}$ is superlinear continuous nonlinearity with subcritical growth at infinity. By using nonsmooth analysis, we investigate the existence and concentration of solutions for the above problem. Moreover, we obtain some properties of these solutions, such as convergence and decay estimate.

The purpose of this paper is to investigate the principal spectral theory and asymptotic behavior of the spectral bound for cooperative nonlocal dispersal systems, specifically focusing on the case where partial diffusion coefficients are zero, referred to as the partially degenerate case. We propose two sufficient conditions that ensure the existence of the principal eigenvalue in these partially degenerate systems. Additionally, we study the asymptotic behavior of the spectral bound for nonlocal dispersal operators with small and large diffusion coefficients, considering both non-degenerate and partially degenerate cases. Notably, we find a threshold-type result as the diffusion coefficients tend towards infinity in the partially degenerate case. Finally, we apply these findings to discuss the asymptotic behavior of the basic reproduction ratio in a viral diffusion model.