The time-global unique classical positive solutions to the reaction–diffusion equations for prey–predator models with dormancy of predators are constructed. The feature appears on the nonlinear terms of Holling type $\rm I\!I$ functional response. The crucial step is to establish time-local positive classical solutions by using a new approximation associated with time-evolution operators. Although the system does not equip usual comparison principle for solutions to partial differential equation, a priori bounds are derived by enclosing and renormalising arguments of solutions to the corresponding ordinary differential equations. Furthermore, time-global existence, invariant regions and asymptotic behaviours of solutions follow from such a priori bounds.

In Chen and Liang previous work, a model, together with its well-posedness, was established for credit rating migrations with different upgrade and downgrade thresholds (i.e. a buffer zone, also called dead band in engineering). When positive dividends are introduced, the model in Chen and Liang (SIAM Financ. Math. 12, 941–966, 2021) may not be well-posed. Here, in this paper, a new model is proposed to include the realistic nonzero dividend scenarios. The key feature of the new model is that partial differential equations in Chen and Liang (SIAM Financ. Math. 12, 941–966, 2021) are replaced by variational inequalities, thereby creating a new free boundary, besides the original upgrading and downgrading free boundaries. Well-posedness of the new model, together with a few financially meaningful properties, is established. In particular, it is shown that when time to debt paying deadline is long enough, the underlying dividend paying company is always in high grade rating, that is, only when time to debt paying deadline is within a certain range, there can be seen the phenomenon of credit rating migration.

We present a stability result for a class of non-trivial self-similarly vanishing solutions to a 3D Hele-Shaw moving boundary problem with surface tension and single-point suction. These solutions are domains that bifurcate from the trivial spherical solution. The moving domains have a geometric centre located at the suction point and they are axially symmetric. We show stability with respect to perturbations that preserve these properties.

The aim of this article is to study the existence and uniqueness of solution for a quasistatic fully coupled thermoelastic problem arising from some metallurgical processes. We consider mixed boundary conditions for both submodels, and a Robin boundary condition for the thermal one. Furthermore, the reference temperature, the thermal conductivity and the Lamé's parameters are assumed to depend on the material point.

The aim of this paper is to develop a theory for the growth of multiple crystals in a polymer melt. This leads to nonlinear moving boundary problems for the temperature, with normal growth speed of the crystal boundaries determined by a nonlinear Gibbs-Thompson relation. Particular attention is paid to the effect of impingement, i.e., the event of two crystals hitting each other, which stops the growth on the contact interface. In one space dimension, the well-posedness of the growth model coupled to the heat equation is shown for an arbitrary number of crystals, both in a quasi-stationary and in an instationary situation. The resulting evolution of a fixed crystal in presence of other crystals is compared to the pure single crystal case. Finally, some basic features of the model in higher spatial dimensions and the main problems encountered in the attempt to prove a general well-posedness result are discussed.

The paper presents the results of an asymptotic theory of axially symmetric cavity flows at small and zero cavitation number. The results have been obtained on the basis of the variational Riabouchinsky principle and the asymptotic theory of slender bodies. This variational-asymptotic approach has been applied to deduce asymptotic expansions for the shape of the cavity and the force exerted on the cavitator at small cavitation number. At the zero cavitation number an integro-differential equation for the shape of the free streamline has been obtained. An exact integral of the equation has been found and a one-parameter family of solutions has been constructed and which has refined earlier asymptotics of Levinson and Gurevich. The equation and asymptotic expansion are independent of the cavitator shape and in this sense are maximally accurate. Any further amendment of the equation of higher order of accuracy would be connected with the shape of the cavitator.

A credit default swap (CDS) is an exchange of premium payments for a compensation for the occurrence of a credit event. Counterparty risks refer to defaults of parties holding CDS contracts. In this paper we develop a valuation/pricing model for a CDS subject to counterparty risks. Using the Cox–Ingersoll–Ross (CIR) model for interest rate and first arrival times of Poisson processes with variable intensities for the occurrences of credit default and counterparty defaults, we derive a mathematical formulation and make a full theoretical investigation. In addition, we develop a full theory for the corresponding infinite horizon problem and establish its connection with the asymptotic long expiry behaviour of finite horizon problem. Furthermore, we establish a connection between two major frameworks for default times: the structure model approach and the intensity model approach. We show that a solution of the structure model can be obtained as the limit of a sequence of solutions of intensity models. Regarded as an important theoretical development, we remove a constraint typically imposed on the parameters of the CIR model; that is, the well-posedness (existence, uniqueness and continuous dependence of parameters) of the mathematical model holds for any empirically calibrated parameters for the CIR model.

Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning (ML) models efficiently. First-order methods such as gradient descent (GD) are usually the methods of choice for training ML models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher et al., arXiv:1806.06317], called modified LSGD (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region’s dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is $\lfloor (n-1)/2\rfloor$, and hence it is significantly smaller than that of GD whose dimension is $n-1$.

We investigate a reaction–diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter $\epsilon$. On the interface between the components, we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk domains are given by a reaction–diffusion equation for the traces of the bulk solutions, including nonlinear reaction kinetics depending on the solutions on both sides of the interface. Using two-scale techniques, we pass to the limit $\epsilon \to 0$ and derive macroscopic models, where we need homogenisation results for surface diffusion. To cope with the nonlinear terms, we derive strong two-scale convergence results.

The Helmholtz equation $-\nabla\cdot (a\nabla u) - \omega^2 u = f$ is considered in an unbounded wave guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$ , $S\subset \mathbb{R}^{d-1}$ a bounded domain. The coefficient a is strictly elliptic and either periodic in the unbounded direction $x_1 \in \mathbb{R}$ or periodic outside a compact subset; in the latter case, two different periodic media can be used in the two unbounded directions. For non-singular frequencies $\omega$ , we show the existence of a solution u. While previous proofs of such results were based on analyticity arguments within operator theory, here, only energy methods are used.

In this paper, we consider a general single population model with delay and patch structure, which could model the population loss during the dispersal. It is shown that the model admits a unique positive equilibrium when the dispersal rate is smaller than a critical value. The stability of the positive equilibrium and associated Hopf bifurcation are investigated when the dispersal rate is small or near the critical value. Moreover, we show the effect of network topology on Hopf bifurcation values for a delayed logistic population model.

We discuss a nonlinear diffusion equation in which the diffusivity has a non-local dependence on the surface concentration. Exact results are obtained for some special cases by means of similarity methods and non-local transformations. We indicate some of the implications the analysis has for more general cases.

In the preceding paper of this issue of EJAM, Bolley & Helffer [5] add to their extensive set of results on bifurcation in the one-dimensional Ginzburg–Landau model of superconductivity. One of their new results concerns the direction of bifurcation of symmetric solutions. We give another approach to this problem.

A class of three-dimensional crack problems is considered, of which a prototype example is provided by a half-space containing a long internal crack, located in a plane perpendicular to the boundary. By means of an asymptotic procedure, the original three-dimensional problem is split up into a sequence of two-dimensional formulations. Results of its numerical implementation are in good agreement with results of more computer-intensive finite-element calculations.

Poroelastic effects have been of great interest in the seismic literature as they have been identified as a major cause of wave attenuation in heterogeneous porous media. The observed attenuation in the seismic wave can be explained in part by energy loss to fluid motion in the pores. On the other hand, it is known that the attenuation is particularly pronounced in stratified structures where the scale of spatial heterogeneity is much smaller than the seismic wavelength. Understanding of poroelastic effects on seismic wave attenuation in heterogeneous porous media has largely relied on numerical experiments. In this work, we present a homogenisation technique to obtain an upscaled viscoelastic model that captures seismic wave attenuation when the sub-seismic scale heterogeneity is periodic. The upscaled viscoelastic model directly relates seismic wave attenuation to the material properties of the heterogeneous structure. We verify our upscaled viscoelastic model against a full poroelastic model in numerical experiments. Our homogenisation technique suggests a new approach for solving coupled equations of motion.

Narrow escape and narrow capture problems which describe the average times required to stop the motion of a randomly travelling particle within a domain have applications in various areas of science. While for general domains, it is known how the escape time decreases with the increase of the trap sizes, for some specific 2D and 3D domains, higher-order asymptotic formulas have been established, providing the dependence of the escape time on the sizes and locations of the traps. Such results allow the use of global optimisation to seek trap arrangements that minimise average escape times. In a recent paper (Iyaniwura (2021) SIAM Rev. 63(3), 525–555), an explicit size- and trap location-dependent expansion of the average mean first passage time (MFPT) in a 2D elliptic domain was derived. The goal of this work is to systematically seek global minima of MFPT for $1\leq N\leq 50$ traps in elliptic domains using global optimisation techniques and compare the corresponding putative optimal trap arrangements for different values of the domain eccentricity. Further, an asymptotic formula for the average MFPT in elliptic domains with N circular traps of arbitrary sizes is derived, and sample optimal configurations involving non-equal traps are computed.

The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen–Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak–strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.

We consider an equation with left and right fractional derivatives which appears as a mathematical model in the mechanics. The type of equations that we analyse appear, as a rule, in variational problems containing fractional derivatives. We look for solutions in a suitably defined sub-space of distributions which is sufficient to enclose different ‘singular’ solutions.

An existence and uniqueness theorem for travelling waves in a bubbly liquid with a continuous bubble-size distribution is proved.

The system of conservation laws governing heat and mass transfer processes in a continuous medium is obtained in a symmetric form on the basis of the successive application of fundamental thermodynamic principles. This approach involves reformulating the problem in intensive thermodynamic variables such as the temperature and chemical potential. The equations of capillary fluid mechanics and phase transitions with moving free boundaries are analysed in detail. The unsteady motion of a drop driven by buoyancy forces in an unbounded ambient fluid with dilute surfactants is investigated where the LeChatelier principle is established for an arbitrary surfactant. The general procedure for construction of self-similar solutions for the thermodiffusive Stefan problem with piecewise constant matrices of coefficients is described