Uniform in Time $L^{\infty }$ -Estimates for Nonlinear Aggregation-Diffusion Equations

In this paper, we consider the general reaction–diffusion system proposed in Abdelmalek and Bendoukha (Nonlinear Anal., Real World Appl. 35:397–413, 2017) as a generalization of the original Lengyel–Epstein model developed for the revolutionary Turing-type CIMA reaction. We establish sufficient conditions for the global existence of solutions. We also follow the footsteps of Lisena (Appl. Math. Comput. 249:67–75, 2014) and other similar studies to extend previous results regarding the local and global asymptotic stability of the system. In the local PDE sense, more relaxed conditions are achieved compared to Abdelmalek and Bendoukha (Nonlinear Anal., Real World Appl. 35:397–413, 2017). Also, new extended results are achieved for the global existence, which when applied to the Lengyel–Epstein system, provide weaker conditions than those of Lisena (Appl. Math. Comput. 249:67–75, 2014). Numerical examples are used to affirm the findings and benchmark them against previous results.

     

Some characterizations of magnetic Sobolev spaces

ABSTRACT

The aim of this note is to survey recent results contained in Nguyen H-M, Squassina M. [On anisotropic Sobolev spaces. Commun Contemp Math, to appear. DOI:10.1142/S0219199718500177]; Nguyen H-M, Pinamonti A, Squassina M, et al. [New characterizations of magnetic Sobolev spaces. Adv Nonlinear Anal. 2018;7(2):227–245]; Pinamonti A, Squassina M, Vecchi E. [Magnetic BV functions and the Bourgain-Brezis-Mironescu formula. Adv Calc Var, to appear. DOI:10.1515/acv-2017-0019]; Pinamonti A, Squassina M, Vecchi E. [The Maz'ya-Shaposhnikova limit in the magnetic setting. J Math Anal Appl. 2017;449:1152–1159] and Squassina M, Volzone B. [Bourgain-Brezis-Mironescu formula for magnetic operators. C R Math Acad Sci Paris. 2016;354:825–831], where the authors extended to the magnetic setting several characterizations of Sobolev and BV functions.     

Cohen positive strongly p-summing and p-convex multilinear operators

The aim of this work is to give and study the notion of Cohen positive p-summing multilinear operators. We prove a natural analog of the Pietsch domination theorem for these classes and characterize their conjugates. As an application, we generalize a result due to Bu and Shi (J. Math. Anal. Appl. 401:174–181, 2013), and we compare this class with the class of multiple p-convex m-linear operators.

     

A New Result on Multiplicity of Nontrivial Solutions for the Nonhomogenous Schrödinger–Kirchhoff Type Problem in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In this paper, we consider the following nonhomogenous Schrödinger–Kirchhoff type problem

$$\left\{ \begin{array}{ll} - (a+b\int_{R^{N}}|\nabla u|^{2}dx)\triangle u + V(x)u =f(x,u)+g(x), & \,\,\,{\rm for} \, x \in R^N, \\ u(x)\rightarrow0, & \,\, {\rm as}\, |x|\rightarrow\infty,\end{array}\right.$$

(0.1)

where constants a > 0, b ≥ 0, N = 1, 2 or 3, \({V\in C(R^{N},R)}\), \({f\in C(R^{N} \times R, R)}\) and \({g\in L^{2}(R^{N})}\). Under more relaxed assumptions on the nonlinear term f that are much weaker than those in Chen and Li (Nonlinear Anal RWA 14:1477–1486, 2013), using some new proof techniques especially the verification of the boundedness of Palais–Smale sequence, a new result on multiplicity of nontrivial solutions for the problem (1.1) is obtained, which sharply improves the known result of Theorem 1.1 in Chen and Li (Nonlinear Anal RWA 14:1477–1486, 2013).

     

Global Weak Solutions for the Cauchy Problem to One-Dimensional Heat-Conductive MHD Equations of Viscous Non-resistive Gas

We investigate the dynamics of the Vlasov-Poisson system in the presence of radiation damping. A propagation result for velocity moments of order \(k>3\) is established in (Kunze and Rendall in Ann. Henri Poincaré 2:857–886, 2001). In this paper, we prove existence of global solutions propagating velocity and velocity-spatial moments of order \(k>2\) and establish an explicit polynomially growing in time bound on the moments.

     

Existence,Nonexistence and Multiplicity Results of a Chern-Simons-Schrödinger System

We study the existence, nonexistence and multiplicity of solutions to Chern-Simons-Schrödinger system

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} -\Delta u+u+\lambda (\frac{h^{2}(|x|)}{|x|^{2}}+\int _{|x|}^{+ \infty }\frac{h(s)}{s}u^{2}(s)ds )u=|u|^{p-2}u,\quad x\in \mathbb{R}^{2}, \\ u\in H^{1}_{r}(\mathbb{R}^{2}), \end{array}\displaystyle \right . \end{aligned}$$

where \(\lambda >0\) is a parameter, \(p\in (2,4)\) and

$$ h(s)=\frac{1}{2} \int _{0}^{s}ru^{2}(r)dr. $$

We prove that the system has no solutions for \(\lambda \) large and has two radial solutions for \(\lambda \) small by studying the decomposition of the Nehari manifold and adapting the fibering method. We also give the qualitative properties about the energy of the solutions and a variational characterization of these extremals values of \(\lambda \). Our results improve some results in Pomponio and Ruiz (J. Eur. Math. Soc. 17:1463–1486, 2015).

     

A $p$ -Laplace Equation with Logarithmic Nonlinearity at High Initial Energy Level

In this paper the authors investigate a class of \(p\)-Laplace equations with logarithmic nonlinearity, which were considered in Le and Le (Acta Appl. Math. 151:149–169, 2017), where, among other things, global existence and finite time blow-up of solutions were proved when the initial energy is subcritical and critical, that is, initial energy smaller than or equal to the depth of the potential well. Their results are complemented in this paper in the sense that an abstract criterion is given for the existence of global solutions that vanish at infinity or solutions that blow up in finite time, when the initial energy is supercritical. As a byproduct it is shown that the problem admits a finite time blow-up solution for arbitrarily high initial energy.

     

On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes

The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and Escobedo (Kinet. Relat. Models, 9(2):251–297, [12]) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namely

$$c(x) = \textstyle\begin{cases} a_{{-}} x \quad x < 1 \\ a_{{+}} x \quad x \geq 1, \end{cases} $$

with \(0 < a_{{+}} < a_{{-}}\). We give necessary and sufficient conditions on the coefficients ensuring the Malthusian behaviour with exponential speed of convergence to an asymptotic profile, and also provide an explicit expression of the latter. Our approach relies crucially on properties of so-called refracted Lévy processes that arise naturally in this setting.

     

An improved local convergence analysis for a two-step Steffensen-type method

We study a class of Steffensen-type algorithm for solving nonsmooth variational inclusions in Banach spaces. We provide a local convergence analysis under ω-conditioned divided difference, and the Aubin continuity property. This work on the one hand extends the results on local convergence of Steffensen’s method related to the resolution of nonlinear equations (see Amat and Busquier in Comput. Math. Appl. 49:13–22, 2005; J. Math. Anal. Appl. 324:1084–1092, 2006; Argyros in Southwest J. Pure Appl. Math. 1:23–29, 1997; Nonlinear Anal. 62:179–194, 2005; J. Math. Anal. Appl. 322:146–157, 2006; Rev. Colomb. Math. 40:65–73, 2006; Computational Theory of Iterative Methods, 2007). On the other hand our approach improves the ratio of convergence and enlarges the convergence ball under weaker hypotheses than one given in Hilout (Commun. Appl. Nonlinear Anal. 14:27–34, 2007).     

Existence and Asymptotic Behavior of Solutions for the Unstirred Chemostat Model with Ratio-Dependent Function

A system of reaction-diffusion equations arising from the unstirred chemostat model with ratio-dependent function is considered. The asymptotic behavior of solutions is given and all positive steady-state solutions to this model lie on a single smooth solution curve. It turns out that the ratio-dependence effect will not affect the dynamics, compared with (Hsu and Waltman in SIAM J. Appl. Math. 53(4):1026–1044, 1993) and (Nie and Wu in Sci. China Math. 56(10):2035–2050, 2013).

     

Exponential Decay of Correlations for a Real-Valued Dynamical System Generated by a $k$ Dimensional System

As a first step towards modelling real time-series, we study a class of real-variable, bounded processes \(\{X_{n}, n\in \mathbb{N}\}\) defined by a deterministic \(k\)-term recurrence relation \(X_{n+k} = \varphi (X _{n}, \ldots , X_{n+k-1})\). These processes are noise-free. We immerse such a dynamical system into \(\mathbb{R}^{k}\) in a slightly distorted way, which allows us to apply the multidimensional techniques introduced by Saussol (Isr. J. Math. 116:223–248, 2000) for deterministic transformations. The hypotheses we need are, most of them, purely analytic and consist in estimates satisfied by the function \(\varphi \) and by products of its first-order partial derivatives. They ensure that the induced transformation \(T\) is dilating. Under these conditions, \(T\) admits a greatest absolutely continuous invariant measure (ACIM). This implies the existence of an invariant density for \(X_{n}\), satisfying integral compatibility conditions. Moreover, if \(T\) is mixing, one obtains the exponential decay of correlations.

     

An Extremal Eigenvalue Problem for a Two-Phase Conductor in a Ball

The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materials. PNLDE 31. Birkhäuser, Basel, 1997) go a long way in showing, in general, that problems of optimal design may not admit solutions if microstructural designs are excluded from consideration. Therefore, assuming, tactilely, that the problem of minimizing the first eigenvalue of a two-phase conducting material with the conducting phases to be distributed in a fixed proportion in a given domain has no true solution in general domains, Cox and Lipton only study conditions for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech. Anal. 136:101–117, 1996). Although, the problem in one dimension has a solution (cf. Kre?n in AMS Transl. Ser. 2(1):163–187, 1955) and, in higher dimensions, the problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino et al. in Nonlinear Anal. TMA 13(2):185–220, 1989), these existence results have been regarded so far as being exceptional owing to complete symmetry. It is still not clear why the same problem in domains with partial symmetry should fail to have a solution which does not develop microstructure and respecting the symmetry of the domain. We hope to revive interest in this question by giving a new proof of the result in a ball using a simpler symmetrization result from Alvino and Trombetti (J. Math. Anal. Appl. 94:328–337, 1983).     

Weighted $$L^2$$ L 2 -norms of Gegenbauer polynomials

We study integrals of the form

$$\begin{aligned} \int _{-1}^1(C_n^{(\lambda )}(x))^2(1-x)^\alpha (1+x)^\beta {{\,\mathrm{\mathrm {d}}\,}}x, \end{aligned}$$

where \(C_n^{(\lambda )}\) denotes the Gegenbauer-polynomial of index \(\lambda >0\) and \(\alpha ,\beta >-1\). We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as \(n\rightarrow \infty \).

     

Effect of the dynamic pressure on the similarity solution of cylindrical shock waves in a rarefied polyatomic gas

The similarity solution for a strong cylindrical shock wave in a rarefied polyatomic gas is analyzed on the basis of Rational Extended Thermodynamics with six independent fields; the mass density, the velocity, the pressure and the dynamic pressure. A new ODE system for the similarity solution is derived in a systematic way by using the method based on the Lie group theory proposed in the context of the spherical shock wave in a rarefied monoatomic gas in Donato and Ruggeri (J Math Anal Appl 251:395, 2000). The boundary conditions are also specified from the Rankine–Hugoniot conditions for the sub-shock. The derived similarity solution is characterized by only one dimensionless parameter \(\alpha \) related to the relaxation time for the dynamic pressure. The numerical analysis of the similarity solution is also performed. The solution agrees with the well-known Sedov–von Neumann–Taylor (SNT) solution when \(\alpha \) is small. When \(\alpha \) is larger, due to the presence of the dynamic pressure, the deviation from the SNT solution is evident; the strength of a peak near the shock front becomes smaller and the profile becomes broader.

     

More good news on the HKM test for multivariate reflected symmetry about an unknown centre

We revisit the problem of testing for multivariate reflected symmetry about an unspecified point. Although this testing problem is invariant with respect to full-rank affine transformations, among the few hitherto proposed tests only a class of tests studied in Henze et al. (J Multivar Anal 87:275–297, 2003) that depends on a positive parameter a respects this property. We identify a measure of deviation \(\varDelta _a\) (say) from symmetry associated with the test statistic \(T_{n,a}\) (say), and we obtain the limit normal distribution of \(T_{n,a}\) as \(n \rightarrow \infty \) under a fixed alternative to symmetry. Since a consistent estimator of the variance of this limit normal distribution is available, we obtain an asymptotic confidence interval for \(\varDelta _a\). The test, when applied to a classical data set, strongly rejects the hypothesis of reflected symmetry, although other tests even do not object against the much stronger hypothesis of elliptical symmetry.

     

Coupled coincidence point theorems in ordered metric spaces

In this paper we extend the coupled contraction mapping theorem proved in partially ordered metric spaces by Gnana Bhaskar and Lakshmikantham (Nonlinear Anal. TMA 65:1379–1393, 2006) to a coupled coincidence point result for a pair of compatible mappings. A control function has been used in our theorem. The mappings are assumed to satisfy a weak contractive inequality. Our theorem improves the results of Harjani et al. (Nonlinear Anal. TMA 74:1749–1760, 2011). The result we have established is illustrated with an example which also shows that the improvement is actual.     

An application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations

In this paper, a class of systems of nonlinear differential equations at resonance is considered. With the use of a global inversion theorem which is an extended form of a non-variational version of a max–min principle, we prove that this class of equations possesses a unique 2π$2\pi$-periodic solution under a rather weaker condition, for existence and uniqueness, than those given in papers [J. Chen, W. Li, Periodic solution for 2k$2k$th boundary value problem with resonance, J. Math. Anal. Appl. 314 (2006) 661–671; F. Cong, Periodic solutions for 2k$2k$th order ordinary differential equations with nonresonance, Nonlinear Anal. 32 (1998) 787–793; F. Cong, Periodic solutions for second order differential equations, Appl. Math. Lett. 18 (2005) 957–961; W. Li, Periodic solutions for 2k$2k$th order ordinary differential equations with resonance, J. Math. Anal. Appl. 259 (2001) 157–167; W. Li, H. Li, A min–max theorem and its applications to nonconservative systems, Int. J. Math. Math. Sci. 17 (2003) 1101–1110; W. Li, Z. Shen, A constructive proof of existence and uniqueness of 2π$2\pi$-periodic solution to Duffing equation, Nonlinear Anal. 42 (2000) 1209–1220]. This result extends the results known so far.     

p-Laplacian problems with nonlinearities interacting with the spectrum

The aim of this paper is investigating the existence and the multiplicity of weak solutions of the quasilinear elliptic problem $$\left\{\begin{array}{ll}-\Delta_p u\ =\ g(x, u) \quad {\rm in} \quad \Omega,\\ u=0 \qquad \qquad \qquad {\rm on}\quad \partial\Omega,\end{array}\right.$$ where ${1 < p < + \infty, \Delta_p u = {\rm div}(|\nabla {u}|^{p-2}\nabla {u})}$ , Ω is an open bounded domain of ${\mathbb{R}^N (N \geq 3)}$ with smooth boundary ?Ω and the nonlinearity g behaves as u ^p?1 at infinity. The main tools of the proof are some abstract critical point theorems in Bartolo et al. (Nonlinear Anal. 7: 981–1012, 1983), but extended to Banach spaces, and two sequences of quasi–eigenvalues for the p–Laplacian operator as in Candela and Palmieri (Calc. Var. 34: 495–530, 2009), Li and Zhou (J. Lond. Math. Soc. 65: 123–138, 2002).