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This paper concerns the bubbling phenomena for the L2$L^2$‐critical half‐wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow‐up solutions concentrating exactly at these singularities. This provides the first examples of multi‐bubble solutions for the half‐wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in Krieger, Lenzmann and Raphaël [Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129] for the single‐bubble case, and explores the localization techniques in Cao, Su and Zhang [Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4] and Röckner, Su and Zhang [Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588] for bubbling solutions to non‐linear Schrödinger equations (NLS). However, unlike the single‐bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control these interactions, as well as non‐local effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half‐wave operator, and find that there exists a narrow room between the orders |t|2+$|t|^{2+}$ and |t|3−$|t|^{3-}$ for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi‐bubble non‐local structure. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of multiplicity of positive solutions to a new generalized Riemann–Liouville type fractional Fisher‐like equation on a semi‐infinite interval equipped with nonlocal multipoint boundary conditions involving Riemann–Liouville fractional derivative and integral operators. The existence of at least two positive solutions for the given problem is established by using the concept of complete continuity and iterative positive solutions. We show the existence of at least three positive solutions to the problem at hand by applying the generalized Leggett–Williams fixed‐point theorem due to Bai and Ge [Z. Bai, B. Ge, Existence of three positive solutions for some second‐order boundary value problems, Comput. Math. Appl. 48 (2014) 699‐70]. Illustrative examples are constructed to demonstrate the effectiveness of the main results. It has also been indicated in Section 5 that some new results appear as special cases by choosing the parameters involved in the given problem appropriately. [ABSTRACT FROM AUTHOR]

In this paper, we focused on a one-dimensional elastic system with a double porosity structure and frictional damping acting on both porous equations. We introduce two stability numbers χ 0 {\chi_{0}} and χ 1 {\chi_{1}} and prove that the solution of the system decays exponentially provided that χ 0 = 0 {\chi_{0}=0} and χ 1 ≠ 0 {\chi_{1}\neq 0} . Otherwise, we prove the absence of exponential decay. Our results improve the results of [N. Bazarra, J. R. Fernández, M. C. Leseduarte, A. Magaña and R. Quintanilla, On the thermoelasticity with two porosities: Asymptotic behaviour, Math. Mech. Solids 24 2019, 9, 2713–2725] and [A. Nemsi and A. Fareh, Exponential decay of the solution of a double porous elastic system, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 83 2021, 1, 41–50]. [ABSTRACT FROM AUTHOR]

In this paper, benefited some ideas of Wang [J. Geom. Anal. 33, 18 (2023)] and Dufera et al. [J. Math. Anal. Appl. 509, 126001 (2022)], we investigate persistence of spatial analyticity for solution of the higher order nonlinear dispersive equation with the initial data in modified Gevrey space. More precisely, using the contraction mapping principle, the bilinear estimate as well as approximate conservation law, we establish the persistence of the radius of spatial analyticity till some time δ. Then, given initial data that is analytic with fixed radius σ0, we obtain asymptotic lower bound σ (t) ≥ c | t | − 1 2 , for large time t ≥ δ. This result improves earlier ones in the literatures, such as Zhang et al. [Discrete Contin. Dyn. Syst. B 29, 937–970 (2024)], Huang–Wang [J. Differ. Equations 266, 5278–5317 (2019)], Liu–Wang [Nonlinear Differ. Equations Appl. 29, 57 (2022)], Wang [J. Geom. Anal. 33, 18 (2023)] and Selberg–Tesfahun [Ann. Henri Poincaré 18, 3553–3564 (2017)]. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of transmission system for coupled Kirchhoff plates, where one equation is conserved and the other has dissipative property, and the dissipation mechanism is given by fractional damping (- Δ) 2 θ v t with θ ∈ [ 1 2 , 1 ] . By using the semigroup method and the multiplier technique, we obtain the exact polynomial decay rates, and find that the polynomial decay rate of the system is determined by the inertia/elasticity ratios and the fractional damping order. Specifically, when the inertia/elasticity ratios are not equal and θ ∈ [ 1 2 , 3 4 ] , the polynomial decay rate of the system is t - 1 / (10 - 4 θ) . When the inertia/elasticity ratios are not equal and θ ∈ [ 3 4 , 1 ] , the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . When the inertia/elasticity ratios are equal, the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . Furthermore it has been proven that the obtained decay rates are all optimal. The obtained results extend the results of Oquendo and Suárez (Z Angew Math Phys 70(3):88, 2019) for the case of fractional damping exponent 2 θ from [0, 1] to [1, 2]. [ABSTRACT FROM AUTHOR]

In this paper, we consider a singular Adams' inequality with logarithmic weights in the unit ball of $ \mathbb {R}^4 $ R 4 . Our results extend the results of Zhu and Wang [Adams' inequality with logarithmic weights in $ \mathbb {R}^4 $ R 4 . Proc Amer Math Soc. 2021;149(8):3463–3472] on Adams' inequality with logarithmic weights to singular case. Then, we study the existence of solutions for some weighted mean field equations, relying on variational methods and the singular Adams' inequality with logarithmic weights we previously established. [ABSTRACT FROM AUTHOR]

The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine p-Laplace nonlocal operator Δ p 풜 , which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math. 386 2021, Article ID 107808] driven by the affine L p energy ℰ p , Ω from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine L p Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive C 1 solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ℰ p , Ω and by the comparison ℰ p , Ω ⁢ (u) ≤ ∥ u ∥ W 0 1 , p ⁢ (Ω) generally strict. [ABSTRACT FROM AUTHOR]

Many articles have recently been devoted to Mahler equations, partly because of their links with other branches of mathematics such as automata theory. Hahn series (a generalization of the Puiseux series allowing arbitrary exponents of the indeterminate as long as the set that supports them is well ordered) play a central role in the theory of Mahler equations. In this paper, we address the following fundamental question: is there an algorithm to calculate the Hahn series solutions of a given linear Mahler equation? What makes this question interesting is the fact that the Hahn series appearing in this context can have complicated supports with infinitely many accumulation points. Our (positive) answer to the above question involves among other things the construction of a computable well‐ordered receptacle for the supports of the potential Hahn series solutions. [ABSTRACT FROM AUTHOR]

By introducing two parameters in the splittings of the matrices A and B, this paper presents a parameterized accelerated iteration (PAI) method for solving the matrix equation A X B = C . The convergence property of the PAI method and the choices of the parameters are thoroughly investigated. Additionally, based on some special splittings of the matrices A and B, several variants of the PAI method are established. Furthermore, for some certain cases, the optimal parameters can be determined, and it is demonstrated that the PAI method is more efficient than the gradient-based iteration (GBI) method (Ding et al. Appl. Math. Comput. 197, 41–50 2008). Finally, by comparing it with several existing iteration methods, the effectiveness of the PAI method is verified through four numerical examples. [ABSTRACT FROM AUTHOR]

We correct the proof of Theorem 2 in the mentioned paper concerning finite-difference equilibrium solutions. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the asymptotic monotonicity of positive solutions for fractional parabolic equation on the right half space. First, a narrow region principle for antisymmetric functions in unbounded domains is obtained, in which we remarkably weaken the decay condition u → 0 at infinity and only assume its growth rate does not exceed | x | γ (0 < γ < 2 s ) compared with (Adv. Math. 377:107463, 2021). Then we obtain asymptotic monotonicity of positive solutions of fractional parabolic equation on R + N × (0 , ∞) . [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of small-amplitude unique solutions near the Maxwellian for the Fokker–Planck–Boltzmann equation in a finite channel with specular reflection boundary conditions. The solution space we consider is denoted as L k ̄ 1 L T ∞ L x 1 , v 2 , introduced in Duan et al. [Commun. Pure Appl. Math. 74(5), 932–1020 (2021)]. In addition, we investigate the long-time behavior of solutions for both hard and soft potentials. [ABSTRACT FROM AUTHOR]

This paper presents some existence and uniqueness results for a solution of a system of equations. Our results extend and generalize the well-known and celebrated results of Boyd and Wong [Proc. Amer. Math. Soc. 20 (1969)], Matkowski [Dissertations Math. (Rozprawy Mat.) 127 (1975)], Proinov [Nonlinear Anal. 64 (2006)], Ri [Indag. Math. (N. S.) 27 (2016)] and many others. We also present some illustrative examples to validate our results. [ABSTRACT FROM AUTHOR]

In the case of oscillatory potentials, we give sufficient conditions for the oscillation of the forced nonlinear second order differential equations with delayed argument in the form x″(t)+q(t)x(τ(t))γsgnx(τ(t))=f(t) in the linear (γ=1) and the superlinear (γ>1) cases. [Copyright &y& Elsevier]

Abstract: In this paper, we present some sixth-order class of modified Ostrowski’s methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixth-order convergent. Several numerical examples are given to illustrate the performance of some of the presented methods. [Copyright &y& Elsevier]

The present paper investigates the following inhomogeneous generalized Hartree equation i u ˙ + Δ u = ± | u | p − 2 | x | b (I α ∗ | u | p | · | b) u , where the wave function is u : = u (t , x) : R × R N → C , with N ≥ 2 . In addition, the exponent b > 0 gives an unbounded inhomogeneous term | x | b and I α ≈ | · | − (N − α) denotes the Riesz-potential for certain 0 < α < N . In this work, our aim is to establish the local existence of solutions in some radial Sobolev spaces, as well as the global existence for small data and the decay of energy sub-critical defocusing global solutions. Our results complement the recent work (Sharp threshold of global well-posedness versus finite time blow-up for a class of inhomogeneous Choquard equations, J. Math. Phys. 60 (2019), 081514). The main challenge in this work is to overcome the singularity of the unbounded inhomogeneous term | x | b for certain b > 0 . [ABSTRACT FROM AUTHOR]

This paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction–repulsion model, with nonlinear productions, diffusion, sensitivities, and logistic term, we detect Lebesgue spaces where given unbounded solutions also blow up in the corresponding norms of those spaces; subsequently, estimates for the blow‐up time are established. Finally, for a simplified version of the model, some blow‐up criteria are proved. More precisely, we analyze a zero‐flux chemotaxis system essentially described as ⋄ut=∇·((u+1)m1−1∇u−χu(u+1)m2−1∇v+ξu(u+1)m3−1∇w)+λu−μukinΩ×(0,Tmax),0=Δv−1|Ω|∫Ωuα+uα=Δw−1|Ω|∫Ωuβ+uβinΩ×(0,Tmax).$$\begin{equation} {\begin{cases} u_t= \nabla \cdot ((u+1)^{m_1-1}\nabla u -\chi u(u+1)^{m_2-1}\nabla v & {}\\ \qquad +\; \xi u(u+1)^{m_3-1}\nabla w) +\lambda u -\mu u^k & \text{ in } \Omega \times (0,T_{max}),\\ 0= \Delta v -\frac{1}{\vert {\Omega }\vert }\int _\Omega u^\alpha + u^\alpha = \Delta w - \frac{1}{\vert {\Omega }\vert }\int _\Omega u^\beta + u^\beta & \text{ in } \Omega \times (0,T_{max}). \end{cases}} \end{equation}$$The problem is formulated in a bounded and smooth domain Ω of Rn$\mathbb {R}^n$, with n≥1$n\ge 1$, for some m1,m2,m3∈R$m_1,m_2,m_3\in \mathbb {R}$, χ,ξ,α,β,λ,μ>0$\chi , \xi , \alpha ,\beta , \lambda ,\mu >0$, k>1$k >1$, and with Tmax∈(0,∞]$T_{max}\in (0,\infty ]$. A sufficiently regular initial data u0≥0$u_0\ge 0$ is also fixed. Under specific relations involving the above parameters, one of these always requiring some largeness conditions on m2+α$m_2+\alpha$, (i)we prove that any given solution to (⋄$\Diamond$), blowing up at some finite time Tmax$T_{max}$ becomes also unbounded in Lp(Ω)$L^{\mathfrak {p}}(\Omega)$‐norm, for all p>n2(m2−m1+α)${\mathfrak {p}}>\frac{n}{2}(m_2-m_1+\alpha)$;(ii)we give lower bounds T (depending on ∫Ωu0p¯$\int _\Omega u_0^{\bar{p}}$) of Tmax$T_{max}$ for the aforementioned solutions in some Lp¯(Ω)$L^{\bar{p}}(\Omega)$‐norm, being p¯=p¯(n,m1,m2,m3,α,β)≥p$\bar{p}=\bar{p}(n,m_1,m_2,m_3,\alpha ,\beta)\ge \mathfrak {p}$;(iii)whenever m2=m3$m_2=m_3$, we establish sufficient conditions on the parameters ensuring that for some u0 solutions to (⋄$\Diamond$) effectively are unbounded at some finite time. Within the context of blow‐up phenomena connected to problem (⋄$\Diamond$), this research partially improves the analysis in Wang et al. (J Math Anal Appl. 2023;518(1):126679) and, moreover, contributes to enrich the level of knowledge on the topic. [ABSTRACT FROM AUTHOR]

This paper studies the nonlocal fractional analog of the famous paper of Brezis and Nirenberg [Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477]. Namely, we focus on the following model: P−Δsu−λu=α|u|p−2u+β|u|2s∗−2uinΩ,u=0inRN∖Ω,$$\begin{align*}\hskip5pc {\left(\mathcal{P}\right)} {\left\{ \def\eqcellsep{&}\begin{array}{l} {\left(-\Delta \right)}^s u-\lambda u = \alpha |u|^{p-2}u + \beta |u|^{2^*_s-2}u \quad \mbox{in}\quad \Omega ,\\ u=0\quad \mbox{in}\quad \mathbb {R}^N\setminus \Omega , \end{array} \right.}\hskip-5pc \end{align*}$$where (−Δ)s$(-\Delta)^s$ is the fractional Laplace operator, s∈(0,1)$s \in (0,1)$, with N>2s$N > 2s$, 20,λ,α∈R$\beta >0,\, \lambda , \alpha \in \mathbb {R}$, and establish the existence of nontrivial solutions and sign‐changing solutions for the problem (P)$(\mathcal{P})$. [ABSTRACT FROM AUTHOR]

In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with Ric ≥ (n − 1) κ , κ = 0 or 1 , which extends the results in [A. Alvino, C. Nitsch and C. Trombetti, A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions, Comm. Pure Appl. Math. 76(3) (2023) 585–603]. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel–Daners inequality for Robin Laplacian. [ABSTRACT FROM AUTHOR]

In this note, we make two revisions of the paper [2]. The first one is the asymptotic behavior of the energy functional as t → T (see [2, Theorem 1.6]), where T is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [2, Theorem 1.8]). [ABSTRACT FROM AUTHOR]

This paper outlines an account of numbers based on the numerical equivalence schema (NES), which consists of all sentences of the form ' # x. F x = n if and only if ∃ n x   F x ', where # is the number-of operator and ∃ n is defined in standard Russellian fashion. In the first part of the paper, I point out some analogies between the NES and the T-schema for truth. In light of these analogies, I formulate a minimalist account of numbers, based on the NES, which strongly parallels the minimalist (deflationary) account of truth. One may be tempted to develop the minimalist account in a fictionalist direction, according to which arithmetic is useful but untrue, if taken at face value. In the second part, I argue that this suggestion is not as attractive as it may first appear. The NES suffers from a similar problem to the T-schema: it is deductively weak and does not enable the derivation of any non-trivial generalizations. In the third part of the paper, I explore some strategies to deal with the generalization problem, again drawing inspiration from the literature on truth. In closing this paper, I briefly compare the minimalist to some other accounts of numbers. [ABSTRACT FROM AUTHOR]

The paper is devoted to the introduction of a new class of generalized set-valued nonlinear variational-like inequality problems in the setting of Banach spaces. By means of the notion of P- η -proximal mapping, we prove its equivalence with a class of generalized implicit Wiener–Hopf equations and employ the obtained equivalence relationship and Nadler's technique to suggest a new iterative algorithm for finding an approximate solution of the considered problem. The existence of solution and the strong convergence of the sequences generated by our proposed iterative algorithm to the solution of our considered problem are verified. The problem of finding a common element of the set of solutions of a generalized nonlinear variational-like inequality problem and the set of fixed points of a total asymptotically nonexpansive mapping is also investigated. The final section deals with the investigation and analysis of the main results appeared in Kazmi and Bhat (Appl Math Comput 166:164–180, 2005) and some comments relating to them are given. The results presented in this article extend and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation x 2 + d y 6 = z p for square-free values 1 ≤ d ≤ 2 0. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, ℚ -curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations. [ABSTRACT FROM AUTHOR]

In this paper, we will show that the Dubrovin–Frobenius prepotentials on the orbit space of the Coxeter group B n constructed by Arsie et al. [Sel. Math. New Ser. 29, 1 (2023)] coincide with the solutions of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations associated with the constrained KP hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we consider equations involving a uniformly elliptic nonlocal operator Aβu(x)=CN,βP.V.∫ℝNa(x−y)(u(x)−u(y))|x−y|N+βdy,$$ {A}_{\beta }u(x)={C}_{N,\beta}\mathrm{P}.\mathrm{V}.\underset{{\mathbb{R}}^N}{\int}\frac{a\left(x-y\right)\left(u(x)-u(y)\right)}{{\left|x-y\right|}^{N+\beta }} dy, $$where the function a:ℝN↦ℝ$$ a:{\mathbb{R}}^N\mapsto \mathbb{R} $$ is uniformly bounded and radial decreasing. We establish some maximum principles for Aβ$$ {A}_{\beta } $$ in bounded and unbounded domains. Since there is no decay condition in the unbounded domain, we make use of an approximate method to estimate the singular integral to get the maximum principle. As applications of these principles, by carrying out the method of moving planes, we give the monotonicity of solutions to the semilinear equation in the coercive epigraph, which extends the result of Dipierro‐Soave‐Valdinoci [Math. Ann.2017, 369(3‐4): 1283–1326]. Moreover, we obtain the radial symmetry and monotonicity of solutions to the generalized Schrödinger equation in a weaker condition, which is the improvement of the result of Tang [Math. Methods Appl. Sci. 2017, 40(7): 2596–2609]. In addition, the maximum principle also plays an important role in acquiring monotonicity of solutions and a Liouville theorem. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem of compressible Navier–Stokes–Vlasov equations under a local alignment regime in a one-dimensional bounded domain. Based on the relative entropy method and compactness argument, we prove that a weak solution of the initial-boundary value problem converges to a strong solution of the limiting two-phase fluid system. This work extends in some sense the previous work of Choi and Jung [Math. Models Methods Appl. Sci. 31(11), 2213–2295 (2021)], which considered the diffusive term ∂ξξfɛ in the kinetic equation. Note that the diffusion term was not considered in this paper. [ABSTRACT FROM AUTHOR]

This paper provides a refinement of the study of asymptotic behaviour for a class of nonlocal in time telegraph equations with positively singular kernels. Based on fundamental properties of relaxation functions and recent representation of the fundamental solution in [Nonlinear Anal. 193 (2020), 111411], we establish the asymptotic expansions of the variance of the stochastic process for both long-time and short-time, which sharply improves the main result in [Proc. Amer. Math. Soc. 149 (2021), 2067–2080] by removing their technical conditions on the regularly varying behaviours and reformulating the asymptotic expansion in a more natural form. By analysing a new noncommutative operation on a subclass of completely positive functions, we provide a new way to construct finitely many ultraslow subdiffusion processes that are rapidly slower than a given ultraslow kernel. Consequently, we show that for a given completely monotonic ultraslow kernel, there is an induced kernel whose corresponding mean square displacement is logarithmic. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a sliding method to investigate the monotonicity of solutions for nonlocal double phase equations. We first derive a narrow region principle in bounded domains. Then we illustrate how to utilize this new method of sliding to obtain monotonicity of solutions for nonlocal double phase equations in bounded domains and the whole space respectively. [ABSTRACT FROM AUTHOR]

The classical methods for stability analysis of structures contain some levels of epistemic uncertainty. This paper presents an alternative method for the analysis of system stability phenomena. The analysis is conducted by using a new method which is called the change of state philosophy. The phenomenon is considered as the change in the state of the system in this method. The basic principle in the formulation is the use of an equation in which the product of the key parameter of the system and its inverse is set equal to 1. Logical reasoning and mathematics principles are used to explicitly derive the basic theory. The results are presented as the Persian curve which is a super function of the state functions. The accuracy of the proposed method has been verified by using several examples related to the ultimate strength analysis of structural systems. The state functions are defined as explicit functions of the state variable. The state variable is an identification parameter of the system. [ABSTRACT FROM AUTHOR]

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [ABSTRACT FROM AUTHOR]

In this paper, as an attempt we consider the linear self-interacting diffusion driven by an α-stable motion, which is the solution to the equation X t α = M t α − θ ∫ 0 t ∫ 0 s (X s α − X r α) d r d s + ν t , where θ ≠ 0 , ν ∈ R and M α is an α-stable motion on R ( 0 < α ≤ 2). The process is an analogue of the self-attracting diffusion (see Durrett-Rogers, Prob. Theory Related Fields92 (1992), 337–349, and Cranston-Le Jan, Math. Ann.303 (1995), 87–93.). The main object of this paper is to prove some limit theorems associated with the solution process X α for 1 2 < α ≤ 2. When θ > 0 we show that ψ α (t) (X t α − X ∞ α) converges to an α-stable random variable in distribution, as t tends to infinity, where ψ α (t) = t 1 / α for 1 ≤ α ≤ 2 and ψ α (t) = t 2 − 1 α for 1 2 < α < 1. When θ < 0 , for all 1 2 < α ≤ 2 we show that, as t → ∞ , J t α (θ , ν , 0) := t e 1 2 θ t 2 X t α converges to ξ ∞ α − ν θ and J t α (θ , ν , n) : = − θ t 2 (J t α (θ , ν , n − 1) − (2 n − 3) ! ! (ξ ∞ α − ν θ)) → (2 n − 1) ! ! (ξ ∞ α − ν θ) a.s. for all n ≥ 1 , where (− 1) ! ! = 1 and ξ ∞ α = ∫ 0 ∞ s e 1 2 θ s 2 d M s α . [ABSTRACT FROM AUTHOR]

This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller [Trans. Amer. Math. Soc., 48 (1940), pp. 488--515], who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference "P. L. Chebyshev -- 200," and it describes the results of their joint studies with Manasa Mandava (1984--2019). [ABSTRACT FROM AUTHOR]

In the present paper, the full range Strichartz estimates for homogeneous Schrödinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz estimates are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the pseudo‐relativistic Hartree equation i∂tψ=−△+m2ψ−1|x|∗|ψ|2ψonℝ3and study travelling solitary waves of the form ψ(t, x) = eitμφ(x − v t) , where v∈ℝ3 denotes travelling velocity. Fröhlich, Jonsson and Lenzmann in [Comm. Math. Phys. 2007, 274:1‐30] proved that for |v|<1 there exists a critical constant Nc(v), such that the travelling waves exist if and only if 0 < N < Nc(v), where N denotes particle number. In this paper, we consider v=(β,0,0) with 0 < β < 1, and let Nc(β)=Nc(v)|v=(β,0,0). We find that Nc(β) is Lipschitz continuity with respect to β. Based on this fact, we then prove that the boosted ground states φβ with ‖φβ‖L22=(1−β)Nc(β) satisfy limβ→0+‖φβ‖H1/2→+∞. The explicit blow‐up profile and rate will be computed. [ABSTRACT FROM AUTHOR]

This paper deals with the following mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity u t − Δ u t − d i v (| ∇ u | p − 2 ∇ u) = | u | q − 2 u log ⁡ | u | in a bounded domain with zero Dirichlet boundary condition, which was studied in our previous paper [J Math Anal Appl. 2019;478(2):393-420]. In view the results of [J Math Anal Appl. 2019;478(2):393-420], for the case (1) 1 < p ≤ q ≤ 2 , i f n ≤ p , ≤ 2 , i f 2 n n + 2 < p < n , < n p n − p , i f p ≤ 2 n n + 2 , the global existence and blow-up results were got when J (u 0) ≤ d , where d denotes the mountain-pass level. But for the case (2) 1 < p ≤ q a n d 2 < q < ∞ , i f n ≤ p , n p n − p , i f 2 n n + 2 < p < n , the blow-up results were got when J (u 0) ≤ M , where M ≤ d is a constant. In this paper, we extend and complete the results of [J Math Anal Appl. 2019;478(2):393-420] on the following three aspects: First, the blow-up results are got when J (u 0) ≤ d and (2) are satisfied. Second, the upper and lower bounds of blow-up time are estimated. Third, the global existence and blow-up results are got when J (u 0) > d. [ABSTRACT FROM AUTHOR]

Following the popularity of the core-EP (c-EP) and weighted core-EP (w-c-EP) inverses, so called one-sided versions of the w-c-EP inverse are introduced recently in Behera et al. (Results Math 75:174 (2020). These extensions are termed as E-w-c-EP and F-w-d-c-EP g-inverses as well as the star E-w-c-EP and the F-w-d-c-EP star classes of g-inverses. The applicability of these g-inverses in solving certain restricted matrix equations has been verified. Several additional results on these classes of g-inverses are established in this paper. In addition, the Moore–Penrose E-w-c-EP inverse and the F-w-d-c-EP Moore–Penrose inverse are proposed using proper expressions that involve the Moore–Penrose inverse and the E-w-c-EP or F-we-d-c-EP inverse. Further, the W-weighted Moore–Penrose c-EP and the W-weighted c-EP Moore–Penrose g-inverses are considered with the aim to extend the considered w-c-EP generalized inverses to rectangular matrices. Characterizations, properties, representations and applications of these inverses are considered. [ABSTRACT FROM AUTHOR]

A challenging open problem in the 3D Hall-MHD theory is to ask whether or not the global weak solutions are smooth. In this paper, we prove that a weak solution is smooth if the diagonal part of the velocity gradient tensor and the non-diagonal part of the magnetic gradient tensor satisfy Ladyzhenskaya–Prodi–Serrin-type conditions. It is physically interesting since the diagonal part of a gradient tensor is related to the deformation while the non-diagonal part is related to the rotation. Moreover, our main theorems improve significantly a criterion in Ye (Comput Math Appl 70(8):2137–2154, 2015) where all entries of the velocity gradient tensor and the magnetic gradient tensor are needed. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^n$$\end{document}. Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite (n-q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n-q)$$\end{document}-moment, where 1q-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit. [ABSTRACT FROM AUTHOR]

The key difference between math as math and math in science is that in science we blend our physical knowledge with our knowledge of math. This blending changes the way we put meaning to math and even the way we interpret mathematical equations. Learning to think about physics with math instead of just calculating involves a number of general scientific thinking skills that are often taken for granted (and rarely taught) in physics classes. In this paper, I give an overview of my analysis of these additional skills. I propose specific tools for helping students develop these skills in subsequent papers. [ABSTRACT FROM AUTHOR]

This paper addresses an article by Felix Klein of 1886, in which he generalized his theory of polynomial equations of degree 5—comprehensively discussed in his Lectures on the Icosahedron two years earlier—to equations of degree 6 and 7. To do so, Klein used results previously established in line geometry. I review Klein's 1886 article, its diverse mathematical background, and its place within the broader history of mathematics. I argue that the program advanced by this article, although historically overlooked due to its eventual failure, offers a valuable insight into a time of crucial evolution of the subject. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which are not well-defined in the classical sense and are understood by using the paracontrolled distribution method introduced in (Gubinelli et al. in Forum Math Pi 3(6):1, 2015). By a new characterization of weighted Hölder spaces and Zvonkin's transformation we prove some new a priori estimates, and therefore establish the global well-posedness for singular HJB equations. As applications, we obtain global well-posedness in polynomial weighted Hölder spaces for KPZ type equations on the real line, as well as modified KPZ equations for which the Cole–Hopf transformation is not applicable. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional p -Laplacian: (− Δ) p α u = f (x , u , ∇ u) , u > 0 in  Ω , u ≡ 0 in  ℝ n ∖ Ω , where Ω is a bounded or an unbounded domain which is convex in x 1 -direction, and (− Δ) p α is the fractional p -Laplacian operator defined by (− Δ) p α u (x) = C n , α , p P. V. ∫ ℝ n | u (x) − u (y) | p − 2 [ u (x) − u (y) ] | x − y | n + α p d y. Under some mild assumptions on the nonlinearity f (x , u , ∇ u) , we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional p -Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math.  335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math.  19(6) (2017) 1750018]. [ABSTRACT FROM AUTHOR]

This paper deals with a Cauchy problem of the inhomogeneous parabolic equation ut=Δu+〈x〉γup+tσw(x) in ℝN×(0,T), where constants γ>0,p>1, and σ>−1. The Japanese brackets 〈x〉γ:=1+|x|2γ; w(≥,≢0) and the initial data are continuous functions in ℝN. We determine the Fujita exponent for global and non‐global solutions of the problem, depending strictly on N,γ and σ, which complete the ones for the nonnegative solutions in J. Math. Anal. Appl. 251 (2000) 624–648 for N=1,2. It is so interesting that the inhomogeneous term leads to the discontinuity of this critical exponent with respect to σ at zero. [ABSTRACT FROM AUTHOR]

A nonoscillation criterion for a second-order linear difference equation is established correcting a result in [1]. [Copyright &y& Elsevier]

This paper presents asymptotic formulas in the case of the following two problems for the Pucci's extremal operators M ± . It is considered the solution u ε (x) of − ε 2 M ± ∇ 2 u ε + u ε = 0 in Ω such that u ε = 1 on Γ. Here, Ω ⊂ R N is a domain (not necessarily bounded) and Γ is its boundary. It is also considered v (x , t) the solution of v t − M ± ∇ 2 v = 0 in Ω × (0 , ∞) , v = 1 on Γ × (0 , ∞) and v = 0 on Ω × { 0 }. In the spirit of their previous works [Berti D, Magnanini R. Asymptotics for the resolvent equation associated to the game-theoretic p-laplacian. Appl Anal. 2019;98(10):1827–1842.; Berti D, Magnanini R. Short-time behavior for game-theoretic p-caloric functions. J Math Pures Appl (9). 2019;(126):249–272.], the authors establish the profiles as ϵ or t → 0 + of the values of u ε (x) and v (x , t) as well as of those of their q-means on balls touching Γ. The results represent a further step in the extensions of those obtained by Varadhan and by Magnanini-Sakaguchi in the linear regime. [ABSTRACT FROM AUTHOR]

In this paper we study a variable-exponent fourth-order viscoelastic equation of the form *** in a bounded domain of Rn. Under suitable conditions on variable exponents and initial data, we prove that the solutions will grow up as an exponential function with positive initial energy level. Our result improves and extends many earlier results in the literature such as the one by Mahdi and Hakem (Ser. Math. Inform. 2020, https://doi.org/10.22190/FUMI2003647M). [ABSTRACT FROM AUTHOR]

We consider the three dimensional free-boundary compressible elastodynamic system under the Rayleigh–Taylor sign condition. This describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor is assumed to satisfy the neo-Hookean linear elasticity. The local well-posedness was proved by Trakhinin (J Differ Eq 264(3):1661–1715, 2018) by Nash–Moser iteration. In this paper, we give a new proof of the local well-posedness by the combination of classical energy method and hyperbolic approach. In the proof, we apply the tangential smoothing method to define the approximation system. The key observation is that the structure of the wave equation of pressure together with Christodoulou–Lindblad (Commun Pure Appl Math 53(12):1536–1602, 2000) elliptic estimates reduces the energy estimates to the control of tangentially-differentiated wave equations despite a potential loss of derivative in the source term. To the best of our knowledge, we first establish the nonlinear energy estimate without loss of regularity for free-boundary compressible elastodynamics. The energy estimate is also uniform in sound speed which yields the incompressible limit, that is, the solutions of the free-boundary compressible elastodynamic equations converge to the incompressible counterpart provided the convergence of initial datum. It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to include the full time derivatives in the boundary energy and analyze higher order wave equations as in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) can still be recovered for a slightly compressible elastic medium by further delicate analysis of the Alinhac good unknowns, which is completely different from Euler equations. [ABSTRACT FROM AUTHOR]

We study the fundamental solutions of parabolic magnetic Schrödinger operators and uniform parabolic operators with nonnegative potentials in the reverse Hölder class. The main aim of the paper is to give pointwise estimates of the heat kernel of the operators above, which improve and generalize the main results by Kurata [J. London Math. Soc. (2) 62 (2000), pp. 885–903]. [ABSTRACT FROM AUTHOR]

In this paper, we propose an embedded low-regularity integrator (ELRI) under a new framework for solving the modified Korteweg-de Vries (mKdV) equation under rough data. Different from the previous work [Wu and Zhao, BIT, Number. Math., (2021)], the present ELRI scheme is constructed based on an approximation of a scaled Schrödinger operator and a new strategy of iterative regularizing through the inverse Miura transform. Moreover, the ELRI scheme is explicitly defined in the physical space, and it is efficient under the Fourier pseudospectral discretization. By rigorous error analysis, we show that ELRI achieves first-order accuracy by requiring the boundedness of one additional spatial derivative of the solution. Numerical results are presented to show the accuracy and efficiency of ELRI. [ABSTRACT FROM AUTHOR]