This paper deals with the existence of solutions to the following system:

$$\left\{\begin{array}{l} -\Deltau+u=\frac{\alpha}{\alpha+\beta}a(x)|v|^{\beta} |u|^{\alpha-2}u\quad\mbox{ in}\mathbb{R}^N\\ [0.2cm] -\Delta v+v=\frac{\beta}{\alpha+\beta}a(x)|u|^{\alpha}|v|^{\beta-2}v\quad\mbox{ in }\mathbb{R}^N. \end{array}\right.$$$\left\{\begin{array}{c}\\ \mathrm{-}\mathit{\Delta u}\mathrm{+}\mathit{u}\mathrm{=}\frac{\mathit{\alpha }}{\mathit{\alpha }\mathrm{+}\mathit{\beta }}\mathit{a}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\left|}\mathit{v}{\mathrm{|}}^{\mathit{\beta }}\mathrm{\right|}\mathit{u}{\mathrm{|}}^{\mathit{\alpha }\mathrm{-}\mathrm{2}}\mathit{u}\mathrm{in}{R}^{\mathit{N}}\\ \mathrm{-}\mathit{\Delta v}\mathrm{+}\mathit{v}\mathrm{=}\frac{\mathit{\beta }}{\mathit{\alpha }\mathrm{+}\mathit{\beta }}\mathit{a}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\left|}\mathit{u}{\mathrm{|}}^{\mathit{\alpha }}\mathrm{\right|}\mathit{v}{\mathrm{|}}^{\mathit{\beta }\mathrm{-}\mathrm{2}}\mathit{v}\mathrm{in}{R}^{\mathit{N}}\mathit{.}\end{array}\right.$

With the help of the Nehari manifold and the linking theorem, we prove the existence ofat least two nontrivial solutions. One of them is positive. Our main tools are theconcentration-compactness principle and the Ekeland’s variational principle.

Time optimal control problems for an internally controlled heat equation with pointwisecontrol constraints are studied. By Pontryagin’s maximum principle and properties ofnontrivial solutions of the heat equation, we derive a bang-bang property for time optimalcontrol. Using the bang-bang property and establishing certain connections between timeand norm optimal control problems for the heat equation, necessary and sufficientconditions for the optimal time and the optimal control are obtained.

A theoretical investigation of the unsteady flow of a Newtonian fluid through a channel is presented using an alternative boundary condition to the standard no-slip condition, namely the Navier boundary condition, independently proposed over a hundred years ago by both Navier and Maxwell. This boundary condition contains an extra parameter called the slip length, and the most general case of a constant but different slip length on each channel wall is studied. An analytical solution for the velocity distribution through the channel is obtained via a Fourier series, and is used as a benchmark for numerical simulations performed utilizing a finite element analysis modified with a penalty method to implement the slip boundary condition. Comparison between the analytical and numerical solution shows excellent agreement for all combinations of slip lengths considered.

Selective withdrawal of a two-layer fluid is considered. The fluid layers are weakly compressible, miscible and viscous and therefore flow rotationally. The lower, denser fluid flows with constant velocity out through one or more drain holes in the bottom of a rectangular tank. The drain is opened impulsively and the subsequent draw-down produces waves in the interface which travel outward to the edges of the tank and are reflected back with a $18{0}^{\circ } $ change of phase. The points on the interface that have the highest absolute gradient form regions of high vorticity in the tank, enabling mixing of the fluids. An inviscid linearized interface is computed and compared to contour plots of density for the viscous solution. The two agree closely at early times in the withdrawal process, but as time increases, nonlinear and viscous effects take over. The time at which the lighter fluid starts to flow out of the tank is dependent on the number of drains, their width, and the fluid flow rate and density, and is investigated here.

We develop a computational method for solving an optimal control problem governed by a switched impulsive dynamical system with time delay. At each time instant, only one subsystem is active. We propose a computational method for solving this optimal control problem where the time spent by the state in each subsystem is treated as a new parameter. These parameters and the jump strengths of the impulses are decision parameters to be optimized. The gradient formula of the cost function is derived in terms of solving a number of delay differential equations forward in time. Based on this, the optimal control problem can be solved as an optimization problem.

We consider a hybrid model, created by coupling a continuum and an agent-based model of infectious disease. The framework of the hybrid model provides a mechanism to study the spread of infection at both the individual and population levels. This approach captures the stochastic spatial heterogeneity at the individual level, which is directly related to deterministic population level properties. This facilitates the study of spatial aspects of the epidemic process. A spatial analysis, involving counting the number of infectious agents in equally sized bins, reveals when the spatial domain is nonhomogeneous.

Increased frequency and severity of stressors associated with climate change are drastically altering ecosystems. Caribbean coral reefs differ markedly from just 30 years ago, with much restructuring attributable to infectious disease outbreaks. Using a classic epidemiological approach, we demonstrate how density-dependent demographic rates serve as a mechanism for intrinsic coral resilience to population perturbations arising from disturbances such as disease. We explore the impact of allowing infection status to influence demographic rates and ascertain outbreak thresholds that are corroborated by epizootic patterns observed in the field. We discuss how our threshold calculations may provide metrics of coral epizootic early warning systems. Integrating our infection model with equations describing the interspecific competition for space between coral and macroalgae, we provide new mechanistic understanding of the influence that coral life history dynamism and infectious disease have on the changing face of these threatened ecosystems.

Random networks were first used to model epidemic dynamics in the 1950s, but in the last decade it has been realized that scale-free networks more accurately represent the network structure of many real-world situations. Here we give an analytical and a Monte Carlo method for approximating the basic reproduction number ${R}_{0} $ of an infectious agent on a network. We investigate how final epidemic size depends on ${R}_{0} $ and on network density in random networks and in scale-free networks with a Pareto exponent of 3. Our results show that: (i) an epidemic on a random network has the same average final size as an epidemic in a well-mixed population with the same value of ${R}_{0} $; (ii) an epidemic on a scale-free network has a larger average final size than in an equivalent well-mixed population if ${R}_{0} \lt 1$, and a smaller average final size than in a well-mixed population if ${R}_{0} \gt 1$; (iii) an epidemic on a scale-free network spreads more rapidly than an epidemic on a random network or in a well-mixed population.

A two-person zero-sum differential game with unbounded controls is considered. Underproper coercivity conditions, the upper and lower value functions are characterized as theunique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacsequations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upperand lower value functions coincide, leading to the existence of the value function of thedifferential game. Due to the unboundedness of the controls, the corresponding upper andlower Hamiltonians grow super linearly in the gradient of the upper and lower valuefunctions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobiequations involving such kind of Hamiltonian is proved, without relying on theconvexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivityconditions guaranteeing the finiteness of the upper and lower value functions are sharp insome sense.

We compute the Γ-limit of a sequence of non-local integral functionalsdepending on a regularization of the gradient term by means of a convolution kernel. Inparticular, as Γ-limit, we obtain free discontinuity functionals withlinear growth and with anisotropic surface energy density.

We consider the following problem of error estimation for the optimal control ofnonlinear parabolic partial differential equations: let an arbitrary admissible controlfunction be given. How far is it from the next locally optimal control? Under naturalassumptions including a second-order sufficient optimality condition for the (unknown)locally optimal control, we estimate the distance between the two controls. To do this, weneed some information on the lowest eigenvalue of the reduced Hessian. We apply thistechnique to a model reduced optimal control problem obtained by proper orthogonaldecomposition (POD). The distance between a local solution of the reduced problem to alocal solution of the original problem is estimated.

The propagation of the action potential in the heart chambers is accurately described bythe Bidomain model, which is commonly accepted and used in the specialistic literature.However, its mathematical structure of a degenerate parabolic system entails highcomputational costs in the numerical solution of the associated linear system. Domaindecomposition methods are a natural way to reduce computational costs, and OptimizedSchwarz Methods have proven in the recent years their effectiveness in accelerating theconvergence of such algorithms. The latter are based on interface matching conditions moreefficient than the classical Dirichlet or Neumann ones. In this paper we analyze anOptimized Schwarz approach for the numerical solution of the Bidomain problem. We assessthe convergence of the iterative method by means of Fourier analysis, and we investigatethe parameter optimization in the interface conditions. Numerical results in 2D and 3D aregiven to show the effectiveness of the method.

We consider the problem of minimising the nth-eigenvalue of the RobinLaplacian in RN. Although for n = 1,2 and apositive boundary parameter α it is known that the minimisers do notdepend on α, we demonstrate numerically that this will not always be thecase and illustrate how the optimiser will depend on α. We derive aWolf–Keller type result for this problem and show that optimal eigenvalues grow at mostwith n1/N, which is in sharp contrast withthe Weyl asymptotics for a fixed domain. We further show that the gap between consecutiveeigenvalues does go to zero as n goes to infinity. Numerical results thensupport the conjecture that for each n there exists a positive value ofαn such that the ntheigenvalue is minimised by n disks for all0 < α < αnand, combined with analytic estimates, that this value is expected to grow withn1/N.

We consider an asymptotic preserving numerical scheme initially proposed by F. Filbet andS. Jin [J. Comput. Phys. 229 (2010)] and G. Dimarco and L.Pareschi [SIAM J. Numer. Anal. 49 (2011) 2057–2077] in thecontext of nonlinear and stiff kinetic equations. Here, we propose a convergence analysisof such a scheme for the approximation of a system of transport equations with a nonlinearsource term, for which the asymptotic limit is given by a conservation law. We investigatethe convergence of the approximate solution (uεh, vεh) to a nonlinear relaxation system, whereε > 0 is a physical parameter andh represents the discretization parameter. Uniform convergence withrespect to ε and h is proved and error estimates arealso obtained. Finally, several numerical tests are performed to illustrate the accuracyand efficiency of such a scheme.

We introduce and analyze a fully-mixed finite element method for a fluid-solidinteraction problem in 2D. The model consists of an elastic body which is subject to agiven incident wave that travels in the fluid surrounding it. Actually, the fluid issupposed to occupy an annular region, and hence a Robin boundary condition imitating thebehavior of the scattered field at infinity is imposed on its exterior boundary, which islocated far from the obstacle. The media are governed by the elastodynamic and acousticequations in time-harmonic regime, respectively, and the transmission conditions are givenby the equilibrium of forces and the equality of the corresponding normal displacements.We first apply dual-mixed approaches in both domains, and then employ the governingequations to eliminate the displacement u of the solid and the pressure pof the fluid. In addition, since both transmission conditions become essential, they areenforced weakly by means of two suitable Lagrange multipliers. As a consequence, theCauchy stress tensor and the rotation of the solid, together with the gradient ofp and the traces of u and p on the boundary of thefluid, constitute the unknowns of the coupled problem. Next, we show that suitabledecompositions of the spaces to which the stress and the gradient of pbelong, allow the application of the Babuška–Brezzi theory and the Fredholm alternativefor analyzing the solvability of the resulting continuous formulation. The unknowns of thesolid and the fluid are then approximated by a conforming Galerkin scheme defined in termsof PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, andcontinuous piecewise linear functions on the boundary. Then, the analysis of the discretemethod relies on a stable decomposition of the corresponding finite element spaces andalso on a classical result on projection methods for Fredholm operators of index zero.Finally, some numerical results illustrating the theory are presented.

The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressiblefluid exhibiting phase transitions between a liquid and a vapor phase in the presence ofcapillarity effects close to phase boundaries. Standard numerical discretizations areknown to violate discrete versions of inherent energy inequalities, thus leading tospurious dynamics of computed solutions close to static equilibria (e.g.,parasitic currents). In this work, we propose a time-implicit discretization of theproblem, and use piecewise linear (or bilinear), globally continuous finite element spacesfor both, velocity and density fields, and two regularizing terms where correspondingparameters tend to zero as the mesh-size h > 0 tends to zero.Solvability, non-negativity of computed densities, as well as conservation of mass, and adiscrete energy law to control dynamics are shown. Computational experiments are providedto study interesting regimes of coefficients for viscosity and capillarity.

In this paper, we first construct a model for free surface flows that takes into accountthe air entrainment by a system of four partial differential equations. We derive it bytaking averaged values of gas and fluid velocities on the cross surface flow in the Eulerequations (incompressible for the fluid and compressible for the gas). The obtained systemis conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation ofthis system to finally construct a two-layer kinetic scheme in which a special treatmentfor the “missing” boundary condition is performed. Several numerical tests on closed waterpipes are performed and the impact of the loss of hyperbolicity is discussed andillustrated. Finally, we make a numerical study of the order of the kinetic method in thecase where the system is mainly non hyperbolic. This provides a useful stability resultwhen the spatial mesh size goes to zero.

A weak solution of the coupling of time-dependent incompressible Navier–Stokes equationswith Darcy equations is defined. The interface conditions include theBeavers–Joseph–Saffman condition. Existence and uniqueness of the weak solution areobtained by a constructive approach. The analysis is valid for weak regularityinterfaces.

A corrector theory for the strong approximation of gradient fields inside periodiccomposites made from two materials with different power law behavior is provided. Eachmaterial component has a distinctly different exponent appearing in the constitutive lawrelating gradient to flux. The correctors are used to develop bounds on the localsingularity strength for gradient fields inside micro-structured media. The bounds aremulti-scale in nature and can be used to measure the amplification of applied macroscopicfields by the microstructure. The results in this paper are developed for materials havingpower law exponents strictly between  −1 and zero.