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In this paper, the existence and the uniqueness of an evolutionary anisotropic $ p_i(x) $ p i (x) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the $ L^{\infty } $ L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the $ L^{\infty } $ L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic $ p(x)- $ p (x) − Laplace equation with convection term via $ L^{\infty }- $ L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc. [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]

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the evolution p, q-Laplacian equations u t = d i v (| ∇ u | p − 2 ∇ u) + u α ∫ Ω v m d x and v t = d i v (| ∇ v | q − 2 ∇ v) + v β ∫ Ω u n d x with 1 (p − 1 − α) (q − 1 − β) , there exist suitable initial data such that vanishing solutions exist. If m n < (p − 1 − α) (q − 1 − β) , we find the explicit scopes of initial data such that the solutions could not vanish, which complete the corresponding classifications of solutions in Math. Methods Appl. Sci. 39 (2016) 1325–1335 and Appl. Math. Comp. 259 (2015) 587–595, respectively. For the critical case m n = (p − 1 − α) (q − 1 − β) , the solutions vanish in finite time with small initial data. [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]

We obtain the global well-posedness and the scattering theory of the solution for the modified Davey–Stewartson system { i ∂ t u + Δ u = | u | 4 u + u v x 1 , (t , x) ∈ R × R 3 , − Δ v = (| u | 2) x 1 in the energy space H 1 ( R 3) in this paper. Since the interaction Morawetz estimate fails and the nonlinearity is non-local, we employ the concentration-compactness argument introduced by Kenig and Merle (Invent Math. 2006;166:645–675) to establish the scattering result. [ABSTRACT FROM AUTHOR]

This paper concerns with the Cauchy problem of the 1-D bipolar hydrodynamic model for semiconductors, a system of Euler-Poisson equations with time-dependent damping effects − J (1 + t) − λ and − K (1 + t) − λ for − 1 < λ < 1. Here, we consider a more physical case that allows the two pressure functions can be different and the doping profile can be non-zero. Different from the previous study [Li HT, Li JY, Mei. M, et al. Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping. J Math Anal Appl. 2019;437:1081-1121] which considered two identical pressure functions and zero doping profile, the asymptotic profiles of the solutions to this model are constant states rather than the nonlinear diffusion waves. When the initial perturbation around the constant states are sufficiently small in the sense of L 2 , by means of the time-weighted energy method, we prove the global existence and uniqueness of the smooth solutions to the Cauchy problem, and obtain the optimal convergence rates of the solutions toward the constant states. [ABSTRACT FROM AUTHOR]

This paper establishes the global well-posedness issue for the full viscous MHD equations in the axisymmetric setting. Global solutions are obtained in critical Besov spaces uniformly to the viscosity when the resistivity is fixed in the spirit of [Abidi H, Hmidi T, Keraani S. On the global well-posedness for the axisymmetric Euler equations. Math. Ann. 2010;347:15–41.], [Hassainia Z. On the global well-posedness of the 3D axisymmetric resistive MHD equations. Ann. Henri Poincaré. 2022;23:2877-2917], [Hmidi T, Zerguine M. Inviscid limit axisymmetric Navier–Stokes system. Differential and Integral Equations. 2009;22(11–12):1223–1246.]. Furthermore, strong convergence in the resolution spaces with a rate of convergence is also studied. [ABSTRACT FROM AUTHOR]

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 consider the regularity criteria of 3D incompressible Boussinesq equations. By using the Littlewood–Paley decomposition technique to establishing a regularity criterion in terms of the one-directional derivative of velocity in Besov spaces. Our result improves some previous works (Liu Q. A regularity criterion for the Navier–Stokes equations in terms of one-directional derivative of the velocity. Acta Appl Math. 2015;140(1):1–9; Gala S, Ragusa M A. On the regularity criterion for the Navier–Stokes equations in terms of one-directional derivative. Asian-Eur J Math. 2017;10(1):1750012). [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time behavior of the chemotaxis-Navier–Stokes system ∂ t n + u ⋅ ∇ n = λ Δ n − ∇ ⋅ (χ (c) n ∇ c) , ∂ t c + u ⋅ ∇ c = μ Δ c − f (c) n , ∂ t u + u ⋅ ∇ u + ∇ P = ζ Δ u − n ∇ φ , ∇ ⋅ u = 0 , t > 0 , x ∈ Ω posed in a strip domain Ω := R 2 × [ 0 , 1 ] ⊂ R 3 . In Peng-Xiang (Math. Models Methods Appl. Sci., 28 (2018), 869-920), the authors have established the global existence of strong solutions to this system with non-flux boundary conditions for n and c and non-slip boundary conditions for u. Our main purpose is to establish the time-decay rates for such solutions. This will be done by using the anisotropic L p interpolation and the iterative techniques. [ABSTRACT FROM AUTHOR]

The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. Applying the Laguerre calculus established on nilpotent Lie groups of step two in Chang et al. [The Laguerre calculus on the nilpotent Lie group of step two. Preprint; 2019. Available from: ], we find the explicit formulas for the heat kernel of sub-Laplace operator and the fundamental solution of power of sub-Laplace operator on nilpotent Lie groups of step two. Calin, Chang and Markina [Generalized Hamilton–Jacobi equation and heat kernel on step two nilpotent Lie groups. In: Gustafsson B, Vasil'ev A, editors. Analysis and mathematical physics. Basel: Birkhüser; 2009 (Trends in mathematics)] also get the formulas for the heat kernel of sub-Laplace operator on nilpotent Lie groups of step two by using the Hamiltonian and Lagrangian formalisms that are related to geometric mechanics. In this paper, we use a totally different method to prove our main results by using the Laguerre calculus, which is more direct from the point of view of Fourier analysis. [ABSTRACT FROM AUTHOR]

One important task in image segmentation is to find a region of interest, which is, in general, a solution of a nonlinear and nonconvex problem. The authors of Chan and Esedoglu (Aspects of total variation regularized L 1 function approximation. SIAM J. Appl. Math. 2005;65:1817–1837) proposed a convex T V − L 1 problem for finding such a region Σ and proved that when a binary input f is given, a solution u ∗ to the convex problem gives rise to other solutions 1 { u ∗ ≥ μ } for a.e. μ ∈ (0 , 1) from which they raised a question of whether or not u ∗ must be binary. The same is to ask if the two-phase Mumford–Shah model in image processing has a unique solution with a binary input. In this paper, we will discuss how to construct a non-binary solution that provides a negative answer to the question through a connection of two ideas, one from the two-phase Mumford–Shah model in image segmentation and the other from mean curvature motions discussed in some geometric problems (e.g. Alter, Caselles, Chambolle. A characterization of convex calibrable sets in R N . Math. Ann. 2005;322:329–366; Chambolle. An algorithm for mean curvature motion. Interfaces Free Boundaries 2004;6:195–218) revealing the nature of non-uniqueness in image segmentation. [ABSTRACT FROM AUTHOR]

We introduce Besov space $ \dot {B}_{p}^{\alpha,q}(\partial \Omega _k) $ B ˙ p α , q (∂ Ω k) and Triebel-Lizorkin space $ \dot {F}^{\alpha,q}_{p}(\partial \Omega _k) $ F ˙ p α , q (∂ Ω k) on a family of model domains $ \partial \Omega _k=\left \{(\mathbf{z},z_{n+1})=(z_1,z_2,\ldots, z_{n+1}):\right. \left. \mbox {Im} (z_{n+1})=\phi (|\mathbf{z}|^2)\right \} $ ∂ Ω k = { (z , z n + 1) = (z 1 , z 2 , ... , z n + 1) : Im (z n + 1) = ϕ (| z | 2) } with $ \phi (x)=x^{k} $ ϕ (x) = x k in $ \mathbf {C}^{n+1} $ C n + 1 which can be considered as a space of homogeneous type in the sense of Coifman RR, Weiss G.[Analyse Harmonique Non-commutative and Certains Espaces Homogǹes. Berlin: Springer; 1971. (Lecture Notes in Math.; 242).], Coifman R, Weiss G. [Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc. 1977;83:569–645.]. We study the sharp estimates on the fundamental solution for the Kohn Laplacian and Cauchy-Szegö projection on $ \partial \Omega _k $ ∂ Ω k in these spaces. [ABSTRACT FROM AUTHOR]

We study the asymptotic expansion of transmission eigenvalues for anisotropic thin layers. We establish a rigorous second-order expansion for simple transmission eigenvalues with respect to the thickness of the layer. The convergence analysis is based on a generalization of Osborn's Theorem to non-linear eigenvalue problems by Moskow [Nonlinear eigenvalue approximation for compact operators. J Math Phys. 2015;56(11):113512]. We also provide formal derivation in more general cases validating the obtained theoretical result. [ABSTRACT FROM AUTHOR]

Motivated by computing medians and means, two proximal splitting methods, backward-backward scheme introduced by Passty [Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces. J Math Anal Appl. 1979;72:383–390.] and barycentric method introduced by Lehdili and Lemaire [Metric spaces of non-positive curvature. Berlin: Springer; 1999.] are investigated in Hadamard space setting. A weak ergodic convergence theorem and some strong convergene results for both schemes are studied. [ABSTRACT FROM AUTHOR]

In this note we reprove the Lipschitz stability for the inverse problem for the Schrödinger operator with finite-dimensional potentials by using quantitative Runge approximation results. This provides a quantification of the Schrödinger version of the argument from Kohn and Vogelius [Determining conductivity by boundary measurements. II. Interior results. Comm Pure Appl Math. 1985;38(5):643–667] and presents a slight variant of the strategy considered in Alessandrini et al. [Lipschitz stability for a piecewise linear Schrödinger potential from local Cauchy data. Asymptotic Anal. 2018;108:115–149] which may prove useful also in the context of more general operators. [ABSTRACT FROM AUTHOR]

Motivated by Duan and Strain [Optimal large-time behavior of the Vlasov–Maxwell–Boltzmann system in the whole space. Comm Pure Appl Math. 2011;64:1497–1546], we prove the global existence of solutions to the Vlasov–Maxwell–Fokker–Planck system near Maxwellians in the whole space by utilizing the refined energy method. In the proof of the global existence, the a priori estimates on the macroscopic component and the electromagnetic field are obtained by use of the macroscopic balance laws and the Maxwell equations, respectively. Moreover, the optimal time decay of solutions to the global Maxwellians, which can be obtained by employing the Fourier analysis. [ABSTRACT FROM AUTHOR]