Showing papers in "Zeitschrift für Angewandte Mathematik und Physik in 2015"

Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis–fluid system

Abstract: The coupled chemotaxis–fluid system $$\left\{ \begin{array}{lll} &n_t + u\cdot abla n = \Delta n - abla \cdot (n abla c) +rn-\mu n^2, \\ & c_t + u\cdot abla c = \Delta c-c+n , \\ & u_t + abla P = \Delta u + n abla \phi + g(x,t), \\ & abla \cdot u = 0, \end{array}\right.$$ is considered under no-flux boundary conditions for n and c and no-slip boundary conditions for u in three-dimensional bounded domains with smooth boundary, where $${r\geq 0}$$ and $${\mu > 0}$$ are given constants and $${\phi\in W^{1, \infty}(\Omega)}$$ and $${g\in C^1(\bar\Omega\times [0, \infty)) \cap L^\infty(\Omega\times (0,\infty))}$$ are prescribed parameter functions. It is shown that under the explicit condition $${\mu\geq 23}$$ and suitable regularity assumptions on the initial data, the corresponding initial-boundary problem possesses a global classical solution which is bounded. Apart from this, it is proved that if r = 0, then both n(·, t) and c(·, t) decay to zero with respect to the norm in $${L^\infty(\Omega)}$$ as $${t\to \infty}$$ , and that if, moreover, $${\int_0^\infty \int_\Omega |g|^2 < \infty}$$ , then also u(·, t)→ 0 in $${L^\infty(\Omega)}$$ as $${t\to \infty}$$ .

Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence

Abstract: In this paper, we study a metamaterial constructed with an isotropic material organized following a geometric structure which we call pantographic lattice. This relatively complex fabric was studied using a continuous model (which we call pantographic sheet) by Rivlin and Pipkin and includes two families of flexible fibers connected by internal pivots which are, in the reference configuration, orthogonal. A rectangular specimen having one side three times longer than the other is cut at 45° with respect to the fibers in reference configuration, and it is subjected to large-deformation plane-extension bias tests imposing a relative displacement of shorter sides. The continuum model used, the presented numerical models and the extraordinary advancements of the technology of 3D printing allowed for the design of some first experiments, whose preliminary results are shown and seem to be rather promising. Experimental evidence shows three distinct deformation regimes. In the first regime, the equilibrium total deformation energy depends quadratically on the relative displacement of terminal specimen sides: Applied resultant force depends linearly on relative displacement. In the second regime, the applied force varies nonlinearly on relative displacement, but the behavior remains elastic. In the third regime, damage phenomena start to occur until total failure, but the exerted resultant force continues to be increasing and reaches a value up to several times larger than the maximum shown in the linear regime before failure actually occurs. Moreover, the total energy needed to reach structural failure is larger than the maximum stored elastic energy. Finally, the volume occupied by the material in the fabric is a small fraction of the total volume, so that the ratio weight/resistance to extension is very advantageous. The results seem to require a refinement of the used theoretical and numerical methods to transform the presented concept into a promising technological prototype.

Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients

Abstract: In the present paper, a two-dimensional solid consisting of a linear elastic isotropic material, for which the deformation energy depends on the second gradient of the displacement, is considered. The strain energy is demonstrated to depend on 6 constitutive parameters: the 2 Lame constants ( $${\lambda}$$ and $${\mu}$$ ) and 4 more parameters (instead of 5 as it is in the 3D-case). Analytical solutions for classical problems such as heavy sheet, bending and flexure are provided. The idea is very simple: The solutions of the corresponding problem of first gradient classical case are imposed, and the corresponding forces, double forces and wedge forces are found. On the basis of such solutions, a method is outlined, which is able to identify the six constitutive parameters. Ideal (or Gedanken) experiments are designed in order to write equations having as unknowns the six constants and as known terms the values of suitable experimental measurements.

Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof

Abstract: Since the works by Gabrio Piola, it has been debated the relevance of higher-gradient continuum models in mechanics. Some authors even questioned the logical consistency of higher-gradient theories, and the applicability of generalized continuum theories seems still open. The present paper considers a pantographic plate constituted by Euler beams suitably interconnected and proves that Piola’s heuristic homogenization method does produce an approximating continuum in which deformation energy depends only on second gradients of displacements. The Γ-convergence argument presented herein shows indeed that Piola’s conjecture can be rigorously proven in a Banach space whose norm is physically dictated by energetic considerations.

Global regularity of 2D generalized MHD equations with magnetic diffusion

Abstract: This paper is concerned with the global regularity of the 2D (two-dimensional) generalized magnetohydrodynamic equations with only magnetic diffusion $${\Lambda^{2\beta} b}$$ . It is proved that when β > 1 there exists a unique global regular solution for this equations. The obtained result improves the previous known ones which require that $${\beta > \frac{3}{2}}$$ . With help of Fourier analysis, Besov spaces and singular integral theory, some delicate estimates on the vorticity $${\omega}$$ and the current j are established to prove our main result.

Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids

Abstract: In this paper, a continuum mixture model with evolving mass densities and porosity is proposed to describe the process of bone remodeling in the presence of bio-resorbable materials as driven by externally applied loads. From a mechanical point of view, both bone tissue and biomaterial are modeled as linear elastic media with voids in the sense of Cowin and Nunziato (J Elast 13:125-147, 1983). In the proposed continuum model, the change of volume fraction related to the void volume is directly accounted for by considering porosity as an independent kinematical field. The bio-mechanical coupling is ensured by the introduction of a suitable stimulus which allows for discriminating between resorption (of both bone and biomaterial) and synthesis (of the sole natural bone) depending on the level of externally applied loads. The presence of a ‘lazy zone’ associated with intermediate deformation levels is also considered in which neither resorption nor synthesis occur. Some numerical solutions of the integro-differential equations associated with the proposed model are provided for the two-dimensional case. Ranges of values of the parameters for which different percentages of biomaterial substitution occur are proposed, namely parameters characterizing initial and maximum values of mass densities of bone tissue and of the bio-resorbable material.

Global boundedness of solutions to a two-species chemotaxis system

Abstract: In this paper, we consider the chemotaxis system of two species which are attracted by the same signal substance $$\left\{\begin{array}{lll}u_t = \Delta u - abla \cdot (u \chi_1(w) abla w) + \mu_1 u(1 - u - a_1 v), \qquad x \in \Omega, \, t >0,\\ v_t = \Delta v - abla \cdot (v \chi_2(w) abla w) + \mu_2 v(1 - a_2u - v),\qquad x \in \Omega, \, t >0,\\ w_t = \Delta w - w + u + v, \qquad \qquad \qquad \qquad \qquad \qquad\,\,\, x \in \Omega,\, t >0 \end{array}\right.$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $${\Omega \subset \mathbb{R}^n}$$ . We prove that if the nonnegative initial data $${(u_0, v_0) \in \big(C^0(\bar{\Omega})\big)^2}$$ and $${w_0 \in W^{1, r}(\Omega)}$$ for some r > n, the system possesses a unique global uniformly bounded solution under some conditions on the chemotaxis sensitivity functions χ 1(w), χ 2(w) and the logistic growth coefficients μ 1, μ 2.

On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects

Abstract: In this paper, we prove some regularity criteria and the local well-posedness of strong solutions to the magnetohydrodynamics with the Hall and ion-slip effects. We also establish global existence and time decay rate for small data.

Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent

Abstract: In this paper, we consider the following semilinear Kirchhoff type equation $$\left\{\begin{array}{ll}-\left(\epsilon^2a + \epsilon b \int \limits_{\mathbb{R}^3}| abla u|^2 \right) \triangle {u}+V(x)u = \mu K(x)|u|^{p-1}u + Q(x)|u|^4u, \,\, \mathrm{in}\, \mathbb{R}^3, \\ u \in H^1(\mathbb{R}^3), \,\, u > 0,\end{array}\right.$$ where \({\epsilon > 0}\) is a small parameter, \({p \in [3,5)}\), a, b are positive constants, μ > 0 is a parameter, and the nonlinear growth of |u|4u reaches the Sobolev critical exponent since 2* = 6 for three spatial dimensions. We prove the existence of a positive ground state solution \({u_\epsilon}\) with exponential decay at infinity for μ > 0 and \({\epsilon}\) sufficiently small under some suitable conditions on the nonnegative functions V, K and Q. Moreover, \({u_\epsilon}\) concentrates around a global minimum point of V as \({\epsilon \rightarrow 0^+}\). The methods used here are based on the concentration-compactness principle of Lions.

Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger–Poisson system in \({\mathbb{R}^3}\)

Abstract: We are interested in the existence and asymptotic behavior of sign-changing solutions to the following nonlinear Schrodinger–Poisson system $$\left\{\begin{array}{ll}-\Delta u+V(x)u+\lambda \phi(x)u =f(u), \ &\quad x \in \mathbb{R}^3,\\ -\Delta \phi=u^2, \ &\quad x \in \mathbb{R}^3,\end{array}\right.$$ where V(x) is a smooth function and λ is a positive parameter. Because the so-called nonlocal term $${\lambda \phi_u(x)u}$$ is involving in the equation, the variational functional of the equation has totally different properties from the case of $${\lambda=0}$$ . Under suitable conditions, combining constraint variational method and quantitative deformation lemma, we prove that the problem possesses one sign-changing solution $${u_\lambda}$$ . Moreover, we show that any sign-changing solution of the problem has an energy exceeding twice the least energy, and for any sequence $${\{\lambda_n\} \rightarrow 0^+(n \rightarrow \infty)}$$ , there is a subsequence $$\{\lambda_{n_k}\}$$ , such that $${u_{\lambda_{n_k}}}$$ converges in $${H^1(\mathbb{R}^3)}$$ to $${u_0}$$ as $${k\rightarrow \infty}$$ , where $${u_0}$$ is a sign-changing solution of the following equation $$-\Delta u+V(x)u=f(u),\quad \ x \in \mathbb{R}^3$$ .

Thick fibrous composite reinforcements behave as special second-gradient materials: three-point bending of 3D interlocks

Abstract: In this paper, we propose to use a second gradient, 3D orthotropic model for the characterization of the mechanical behavior of thick woven composite interlocks. Such second-gradient theory is seen to directly account for the out-of-plane bending rigidity of the yarns at the mesoscopic scale which is, in turn, related to the bending stiffness of the fibers composing the yarns themselves. The yarns’ bending rigidity evidently affects the macroscopic bending of the material and this fact is revealed by presenting a three-point bending test on \({0 ^{\circ}/90 ^{\circ} \,\,{\rm and}\,\, \pm45 ^{\circ}}\) specimens of composite interlocks. These specimens differ one from the other for the different relative direction of the yarns with respect to the edges of the sample itself. Both types of specimens are independently seen to take advantage of a second-gradient modeling for the correct description of their macroscopic bending modes. The results presented in this paper are essential for the setting up of a correct continuum framework suitable for the mechanical characterization of composite interlocks. The few second-gradient parameters introduced by the present model are all seen to be associated with peculiar deformation modes of the mesostructure (bending of the yarns) and are determined by inverse approach. Although the presented results undoubtedly represent an important step toward the complete characterization of the mechanical behavior of fibrous composite reinforcements, more complex hyperelastic second-gradient constitutive laws must be conceived in order to account for the description of all possible mesostructure-induced deformation patterns.

A microstructure- and surface energy-dependent third-order shear deformation beam model

Abstract: A new non-classical third-order shear deformation model is developed for Reddy–Levinson beams using a variational formulation based on Hamilton’s principle. A modified couple stress theory and a surface elasticity theory are employed. The equations of motion and complete boundary conditions for the beam are obtained simultaneously. The new model contains a material length scale parameter to account for the microstructure effect and three surface elastic constants to describe the surface energy effect. Also, Poisson’s effect is incorporated in the new beam model. The current non-classical model recovers the classical elasticity-based third-order shear deformation beam model as a special case when the microstructure, surface energy and Poisson’s effects are all suppressed. In addition, the newly developed beam model includes the models considering the microstructure dependence or the surface energy effect alone as limiting cases and reduces to two existing models for Bernoulli–Euler and Timoshenko beams incorporating the microstructure and surface energy effects. To illustrate the new model, the static bending and free vibration problems of a simply supported beam loaded by a concentrated force are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that both the deflection and rotation of the simply supported beam predicted by the current model are smaller than those predicted by the classical model. Also, it is observed that the differences in the deflection and rotation predicted by the two beam models are very large when the beam thickness is sufficiently small, but they are diminishing with the increase in the beam thickness. For the free vibration problem, it is found that the natural frequency predicted by the new model is higher than that predicted by the classical beam model, and the difference is significant for very thin beams. These predicted trends of the size effect at the micron scale agree with those observed experimentally.

The Nehari manifold for nonlocal elliptic operators involving concave-convex nonlinearities

Abstract: In this paper, we study the multiplicity of solutions to equations driven by a nonlocal integro-differential operator $${{\mathcal{L}}_K}$$ with homogeneous Dirichlet boundary conditions. In particular, using fibering maps and Nehari manifold, we obtain multiple solutions to the following fractional elliptic problem $$\left\{\begin{array}{ll}(-\triangle)^su(x)=\lambda u^q+ u^p,\quad u > 0 \; {\rm in}\; \Omega;\\ u=0, \qquad\qquad\qquad\qquad\quad \,\,\,{\rm in}\; {\mathbb{R}}^N\backslash\Omega,\end{array}\right.$$ where Ω is a smooth bounded set in $${{\mathbb{R}}^n}$$ , n > 2s with $${s \in (0,1)}$$ , λ is a positive parameter, the exponents p and q satisfy $${0 < q < 1 < p\; \leqslant \; 2_s^\ast-1}$$ with $${2_s^\ast=\frac{2n}{n-2s}}$$ .

Existence and energy decay of solutions for the Euler–Bernoulli viscoelastic equation with a delay

Abstract: In this paper, we consider initial-boundary value problem of Euler–Bernoulli viscoelastic equation with a delay term in the internal feedbacks. Namely, we study the following equation $$u_{tt}(x,t)+ \Delta^2 u(x,t)-\int\limits_0^t g(t-s)\Delta^2 u(x,s){\rm d}s+\mu_1u_t(x,t)+\mu_2 u_t(x,t-\tau)=0 $$ together with some suitable initial data and boundary conditions in $${\Omega\times (0,+\infty)}$$ . For arbitrary real numbers μ 1 and μ 2, we prove that the above-mentioned model has a unique global solution under suitable assumptions on the relaxation function g. Moreover, under some restrictions on μ 1 and μ 2, exponential decay results of the energy for the concerned problem are obtained via an appropriate Lyapunov function.

Boundedness in a quasilinear fully parabolic Keller–Segel system with logistic source

Abstract: This paper deals with the Neumann boundary value problem for the system $$\left\{\begin{array}{lll}u_t = abla \cdot \left(D(u) abla u\right) - abla \cdot \left(S(u) abla v\right) + f(u), &\quad x \in \Omega, \, t > 0,\\ v_t = \Delta v - v + u, &\quad x \in \Omega, \, t > 0\end{array}\right.$$ in a smooth bounded domain $${\Omega\subset{\mathbb{R}}^n}$$ $${(n\geq1)}$$ , where the functions D(u) and S(u) are supposed to be smooth satisfying $${D(u)\geq Mu^{-\alpha}}$$ and $${S(u)\leq Mu^{\beta}}$$ with M > 0, $${\alpha\in{\mathbb{R}}}$$ and $${\beta\in{\mathbb{R}}}$$ for all $${u\geq1}$$ , and the logistic source f(u) is smooth fulfilling $${f(0)\geq0}$$ as well as $${f(u)\leq a-\mu u^{\gamma}}$$ with $${a\geq0}$$ , $${\mu > 0}$$ and $${\gamma\geq1}$$ for all $${u\geq0}$$ . It is shown that if $$\alpha + 2\beta < \left\{\begin{array}{lll}\gamma - 1 + \frac{2}{n}, &\quad {\rm for} \, 1 \leq \gamma < 2,\\ \gamma - 1 + \frac{4}{n + 2}, &\quad {\rm for} \, \gamma \geq 2,\end{array}\right.$$ then for sufficiently smooth initial data, the problem possesses a unique global classical solution which is uniformly bounded.

Shape sensitivity analysis of the energy integrals for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion

Abstract: An equilibrium problem for an elastic Timoshenko-type plate containing a rigid inclusion is considered. On the interface between the elastic plate and the rigid inclusion, there is a vertical crack. Inequality-type boundary conditions are imposed at the crack faces to guarantee mutual nonpenetration. By using a sufficiently smooth perturbation determined in the middle plate plane, the variation of plate geometry is specified. The formula of the derivative of the plate energy functional with respect to the perturbation parameter is deduced.

Equilibrium configurations and stability of a damaged body under uniaxial tractions

Abstract: This paper deals with the equilibrium problem in nonlinear dissipative inelasticity of damaged bodies subject to uniaxial loading. To model the damage effects, a damage function, affecting the stored energy function, is defined. In the framework of the continuum thermodynamics theory, the constitutive law for damaged hyperelastic materials and an inequality for the energy release rate are derived. By means of an energy-based damage criterion, the irreversible evolution law for the damage function is obtained. After formulating the equilibrium boundary value problem, explicit expressions governing the global development of the equilibrium paths are written. Successively, the stability of the equilibrium solutions are assessed through the energy criterion. For a damaged body under uniaxial loading, seven inequalities are derived. These conditions, if fulfilled, ensure the stability of the solutions under each type of small perturbation. Finally, a number of applications for compressible neo-Hookean and Mooney–Rivlin materials are performed.

Well-posedness results for the short pulse equation

Abstract: The short pulse equation provides a model for the propagation of ultra-short light pulses in silica optical fibers. It is a nonlinear evolution equation. In this paper, the well-posedness of bounded solutions for the homogeneous initial boundary value problem and the Cauchy problem associated with this equation are studied.

The existence of normalized solutions for L 2 -critical constrained problems related to Kirchhoff equations

Abstract: In this paper, we study the existence of critical points for the following functional constrained on $${S_c=\{u\in H^1(\mathbb{R}^N)| |u|_2=c\}}$$ : $$I(u)=\frac{a}{2}\int_{\mathbb{R}^N}| abla{u}|^{2}+\frac{b}{4}\left(\int_{\mathbb{R}^N}| abla{u}|^{2}\right)^{2}-\frac{N}{2N+8}\int_{\mathbb{R}^N}|u|^{\frac{2N+8}{N}},$$ where N = 1, 2, 3 and a, b > 0 are constants. The constraint problem is L 2-critical. We showed that I(u) has a constraint critical point with a mountain pass geometry on S c if $${c > c^*:=(2^{-1}b|Q|_2^{\frac{8}{N}})^{\frac{N}{8-2N}}}$$ , where Q is the unique positive radial solution of $${-2\Delta Q+(\frac{4}{N}-1)Q=|Q|^{\frac{8}{N}} Q}$$ in $${\mathbb{R}^N}$$ . For 0 < c < c *, I(u) has no critical point on S c , and we proved the existence of minimizers for a new perturbation functional on S c : $$E_{a,b}(u)=\frac{a}{2} \int_{\mathbb{R}^N}| abla u|^2+\frac{b}{4} \left(\int_{\mathbb{R}^N}| abla u|^2\right)^2-\frac{1}{4} \int_{\mathbb{R}^N}V(x)|u|^{4}-\frac{N}{2N+8} \int_{\mathbb{R}^N}|u|^{\frac{2N+8}{N}}.$$

The exponentiated Hencky-Logarithmic strain energy: Part II: Coercivity, planar polyconvexity and existence of minimizers

Abstract: We consider a family of isotropic volumetric–isochoric decoupled strain energies $$F \mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll}\frac{\mu}{k}\,e^{k\,\|{\rm dev}_n{\rm log} {U}\|^2}+\frac{\kappa}{2\hat{k}}\,e^{\hat{k}\,[{\rm tr}({\rm log} U)]^2}&\text{if}& \det\, F > 0,\\+\infty &\text{if} &\det F\leq 0,\end{array}\right.$$ based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, \({\kappa=\frac{2\mu+3\lambda}{3} > 0}\) is the infinitesimal bulk modulus with \({\lambda}\) the first Lame constant, \({k,\hat{k}}\) are dimensionless parameters, \({F= abla \varphi}\) is the gradient of deformation, \({U=\sqrt{F^T F}}\) is the right stretch tensor and \({{\rm dev}_n{\rm log} {U} ={\rm log} {U}-\frac{1}{n}{\rm tr}({\rm log} {U})\cdot{1\!\!1}}\) is the deviatoric part (the projection onto the traceless tensors) of the strain tensor log U. For small elastic strains, the energies reduce to first order to the classical quadratic Hencky energy $$\begin{array}{ll}F\mapsto W{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U)&:={\mu}\,\|{\rm dev}_n{\rm log} U\|^2+\frac{\kappa}{2}\,[{\rm tr}({\rm log} U)]^2,\end{array}$$ which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family \({W_{_{\rm eH}}}\) are polyconvex for \({k\geq \frac{1}{3},\,\widehat{k}\geq \frac{1}{8}}\) , extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann’s polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor U. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations, and we prove the existence of minimizers by the direct methods of the calculus of variations.

Shape derivative of the energy functional in a problem for a thin rigid inclusion in an elastic body

Abstract: The equilibrium problem of the elastic body with a delaminated thin rigid inclusion is considered. In this case, there is a crack between the rigid inclusion and the elastic body. We suppose that the nonpenetration conditions are prescribed on the crack faces. We study the dependence of the energy of the body on domain variations. The formula for the shape derivative of the energy functional is obtained. Moreover, it is shown that for the special cases of the domain perturbations such derivative can be represented as invariant integrals.

A shear-shear torsional beam model for nonlinear aeroelastic analysis of tower buildings

Abstract: In this paper, an equivalent one-dimensional beam model immersed in a three-dimensional space is proposed to study the aeroelastic behavior of tower buildings: linear and nonlinear dynamics are analyzed through a simple but realistic physical modeling of the structure and of the load. The beam is internally constrained, so that it is capable to experience shear strains and torsion only. The elasto-geometric and inertial characteristics of the beam are identified from a discrete model of three-dimensional frame, via a homogenization process. The model accounts for the torsional effect induced by the rotation of the floors around the tower axis; the macroscopic shear strain is produced by bending of the columns, accompanied by negligible rotation of the floors. Nonlinear aerodynamic forces are evaluated through the quasi-steady theory. The first aim is to investigate the effect of mechanical and aerodynamic coupling on the critical galloping conditions. Furthermore, the role of aerodynamic nonlinearities on the galloping post-critical behavior is analyzed through a perturbation solution which permits to obtain a reduced one-dimensional dynamical system, capable of capturing the essential dynamics of the problem.

Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

Abstract: This paper deals with an initial-boundary value problem for the chemotaxis system $$\left\{\begin{array}{ll} u_t = abla \cdot (D (u) abla u)- abla \cdot (u abla v), \quad & x\in \Omega, \quad t > 0, \\ v_t= \Delta v-uv, \quad & x \in \Omega, \quad t > 0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a convex smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) with \({n\geq3}\), where the diffusion function D(u) satisfying $$\begin{array}{ll}D(u)\geq c_Du^{m-1}\quad\text{for all}\,\,u > 0 \end{array}$$ with some cD > 0 and m > 1. The main goal of this paper was to extend a previous result on global existence of solutions by Wang et al. (Z Angew Math Phys 65:1137–1152, 2014) under the condition that \({m > 2-\frac{2}{n}}\) can be relaxed to \({m > 2-\frac{6}{n+4}}\).

On the nonlinear Schrödinger–Poisson systems with sign-changing potential

Abstract: In this paper, we study a nonlinear Schrodinger–Poisson system $$\left\{ \begin{array}{ll} -\Delta u+V_{\lambda }( x) u+\mu K( x) \phi u=f (x, u) & \text{in}\;\mathbb{R}^{3}, \\ -\Delta \phi =K ( x ) u^{2} & \text{in}\;\mathbb{R}^{3},\end{array}\right.$$ where \({\mu > 0}\) is a parameter, \({V_{\lambda }}\) is allowed to be sign-changing and f is an indefinite function. We require that \({V_{\lambda }:=\lambda V^{+}-V^{-}}\) with V+ having a bounded potential well Ω whose depth is controlled by λ and \({V^{-} \geq 0}\) for all \({x\in \mathbb{R} ^{3}}\). Under some suitable assumptions on K and f, the existence and the nonexistence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is explored as well.

Poisson problems for semilinear Brinkman systems on Lipschitz domains in {\mathbb{R}^n}

Abstract: The purpose of this paper is to combine a layer potential analysis with the Schauder fixed point theorem to show the existence of solutions of the Poisson problem for a semilinear Brinkman system on bounded Lipschitz domains in $${{\mathbb R}^n (n\geq 2)}$$ with Dirichlet or Robin boundary conditions and data in L 2-based Sobolev spaces. We also obtain an existence and uniqueness result for the Dirichlet problem for a special semilinear elliptic system, called the Darcy–Forchheimer–Brinkman system.

Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity

Abstract: In this paper, we consider an initial boundary value problem for the magnetohydrodynamic compressible flows. By assuming that the heat conductivity depends on temperature with κ (θ) = θ q , q > 0, we prove the existence and uniqueness of global strong solutions with large initial data and show that neither shock waves nor vacuum and concentration of mass in the solutions are developed in a finite time.

Limit cycles for m-piecewise discontinuous polynomial Lienard differential equations

Abstract: We provide lower bounds for the maximum number of limit cycles for the m-piecewise discontinuous polynomial differential equations $${\dot{x} = y+{\rm sgn}(g_m(x, y))F(x)}$$ , $${\dot{y} = -x}$$ , where the zero set of the function sgn(g m (x, y)) with m = 2, 4, 6, . . . is the product of m/2 straight lines passing through the origin of coordinates dividing the plane into sectors of angle 2π/m, and sgn(z) denotes the sign function.

A mode III arc-shaped crack with surface elasticity

Abstract: We study the effect of surface elasticity on an arc-shaped crack in a linearly elastic isotropic homogeneous material under antiplane shear deformation. The surface mechanics is incorporated by using a continuum-based surface/interface model of Gurtin and Murdoch. We obtain a complete solution by reducing the problem to two decoupled first-order Cauchy-type singular integro-differential equations. It is shown that different from the case of a straight crack, the stresses exhibit both the weak logarithmic and the strong square root singularities at the tips of the arc crack.

Well-posedness of the modified Camassa–Holm equation in Besov spaces

Abstract: In this paper, we consider the modified Camassa–Holm equation of the form $$y_t + 2 u_x y + uy_x = 0, \quad y = (1 - \partial_x^2)^{2}u.$$ We prove that the Cauchy problem for this equation is locally well-posed in the critical Besov space \({B_{2, 1}^{7/2}}\) or in \({B_{p, r}^{s}}\) with \({1\leq p, r\leq + \infty}\), \({s > \max\{3 + 1/p, 7/2\}}\). Particularly, our method used to prove the local well-posedness in \({B_{2, 1}^{7/2}}\) is different from the previous one used in critical Besov space which involves extracting a convergent subsequence from an iterative sequence. We also prove that if a weaker \({B_{p, r}^q}\)-topology is used, then the solution map becomes Holder continuous. Furthermore, we obtain the peakon-like solution which enable us to prove the ill-posedness in \({B_{2, \infty}^{7/2}}\). Finally, when \({x \in \mathbb{T} = \mathbb{R}/2 \pi \mathbb{Z}}\), we show that the solution map is not uniformly continuous in \({B_{2, r}^{s}}\) with \({1\leq r\leq \infty}\) and \({s > 7/2}\) or \({r = 1, s = 7/2}\).

Singular phenomena of solutions for nonlinear diffusion equations involving p(x)-Laplace operator and nonlinear sources

Abstract: The aim of this paper was to study vanishing and blowing-up properties of the solutions to a homogeneous initial Dirichlet problem of a nonlinear diffusion equation involving the p(x)-Laplace operator and a nonlinear source. The authors point out that the results obtained are not trivial generalizations of similar problems in the case of constant exponent because the variable exponent p(x) brings some essential difficulties such as the failure of upper and lower solution method and scaling technique, the existence of a gap between the modular and the norm. To overcome these difficulties, the authors have to improve the regularity of solutions, to construct a new control functional and apply suitable embedding theorems to prove the blowing-up property of the solutions. In addition, the authors utilize an energy estimate method and a comparison principle for ODE to prove that the solution vanishes in finite time. At the same time, the critical extinction exponents and an extinction rate estimate to the solutions are also obtained.