Showing papers in "Nonlinearity in 2018"
TL;DR: In this article, general soliton solutions to nonlinear Schrodinger (NLS) with Parity (PT)-symmetry for both zero and nonzero boundary conditions are obtained.
Abstract: General soliton solutions to a nonlocal nonlinear Schrodinger (NLS) equation with PT-symmetry for both zero and nonzero boundary conditions are considered via the combination of Hirota's bilinear method and the Kadomtsev–Petviashvili (KP) hierarchy reduction method. First, general N-soliton solutions with zero boundary conditions are constructed. Starting from the tau functions of the two-component KP hierarchy, it is shown that they can be expressed in terms of either Gramian or double Wronskian determinants. On the contrary, from the tau functions of single component KP hierarchy, general soliton solutions to the nonlocal NLS equation with nonzero boundary conditions are obtained. All possible soliton solutions to nonlocal NLS with Parity (PT)-symmetry for both zero and nonzero boundary conditions are found in the present paper.
134 citations
TL;DR: In this article, a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator is studied, and it is shown that the local nonnegative solutions blowup in finite time with arbitrary negative initial energy and suitable initial values.
Abstract: In this paper, we study a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator. As a particular case, we consider the following diffusion problem where [u] s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, is the fractional Laplacian with , is the initial function, and is continuous. Under some appropriate conditions, the local existence of nonnegative solutions is obtained by employing the Galerkin method. Then, by virtue of a differential inequality technique, we prove that the local nonnegative solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give an estimate for the lower and upper bounds of the blow-up time. The main novelty is that our results cover the degenerate case, that is, the coefficient of could be zero at the origin.
98 citations
TL;DR: In this article, the authors considered radially symmetric solutions of the Keller-Segel system with nonlinear signal production, where f is a suitably regular function generalizing the prototype determined by the choice,, with.
Abstract: This paper is concerned with radially symmetric solutions of the Keller–Segel system with nonlinear signal production, as given by in the ball for and R > 0, where f is a suitably regular function generalizing the prototype determined by the choice , , with . The main results assert that if in this setting the number κ satisfies then for any prescribed mass level m > 0, there exist initial data u 0 with , for which the solution of the corresponding Neumann initial-boundary value problem blows up in finite time. The condition in () is essentially optimal and is indicated by a complementary result according to which in the case , for widely arbitrary initial data, a global bounded classical solution can always be found.
90 citations
TL;DR: In this article, Fokas presented a nonlocal nonlinear Schrodinger (NLS) equation with a self-induced parity-time-symmetric potential, which is a two-spatial dimensional analogue of the nonlinear nonlinear NLS equation.
Abstract: Recently, Fokas presented a nonlocal Davey–Stewartson I (DSI) equation (Fokas 2016 Nonlinearity 29 319–24), which is a two-spatial dimensional analogue of the nonlocal nonlinear Schrodinger (NLS) equation (Ablowitz and Musslimani 2013 Phys. Rev. Lett. 110 064105), involving a self-induced parity-time-symmetric potential. For this equation, high-order periodic line waves and line breathers are derived by employing the bilinear method. The long wave limit of these periodic solutions yields two kinds of fundamental rogue waves, namely, kink-shaped and W-shaped line rogue waves. The interaction of fundamental line rogue waves generate higher-order rogue waves, which have several interesting patterns with different curvy profiles. Furthermore, semi-rational solutions are constructed, which are line rogue waves on a background of periodic line waves. Finally, two particular solutions of the nonlocal NLS equation, namely, a first-order rogue wave and a semi-rational solution are obtained as reductions of the corresponding solutions of the nonlocal DSI equation.
86 citations
TL;DR: In this article, two families of travelling periodic waves of the modified Korteweg-de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn.
Abstract: Rogue periodic waves stand for rogue waves on a periodic background. Two families of travelling periodic waves of the modified Korteweg–de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn. By using one-fold and two-fold Darboux transformations of the travelling periodic waves, we construct new explicit solutions for the mKdV equation. Since the dn-periodic wave is modulationally stable with respect to long-wave perturbations, the new solution constructed from the dn-periodic wave is a nonlinear superposition of an algebraically decaying soliton and the dn-periodic wave. On the other hand, since the cn-periodic wave is modulationally unstable with respect to long-wave perturbations, the new solution constructed from the cn-periodic wave is a rogue wave on the cn-periodic background, which generalizes the classical rogue wave (the so-called Peregrine's breather) of the nonlinear Schrodinger equation. We compute the magnification factor for the rogue cn-periodic wave of the mKdV equation and show that it remains constant for all amplitudes. As a by-product of our work, we find explicit expressions for the periodic eigenfunctions of the spectral problem associated with the dn and cn periodic waves of the mKdV equation.
83 citations
TL;DR: In this article, it was shown that the ground states of the Gross-Pitaevskii (GP) equation are unique and radially symmetric at least for almost every.
Abstract: We are interested in the attractive Gross–Pitaevskii (GP) equation in , where the external potential vanishes on m disjoint bounded domains and as , that is, the union of these is the bottom of the potential well. By establishing some delicate estimates on the associated energy functional of the GP equation, we prove that when the interaction strength a approaches some critical value , the ground states concentrate and blow up at the center of the incircle of some with the largest inradius. Moreover, under some further conditions on , we show that the ground states of the GP equations are unique and radially symmetric at least for almost every .
65 citations
TL;DR: In this paper, the existence of multiple positive solutions to the nonlinear Schrodinger systems set on, under the constraint Here are prescribed,, and the frequencies are unknown and will appear as Lagrange multipliers.
Abstract: We consider the existence of multiple positive solutions to the nonlinear Schrodinger systems set on , under the constraint Here are prescribed, , and the frequencies are unknown and will appear as Lagrange multipliers. Two cases are studied, the first when , the second when In both cases, assuming that is sufficiently small, we prove the existence of two positive solutions. The first one is a local minimizer for which we establish the compactness of the minimizing sequences and also discuss the orbital stability of the associated standing waves. The second solution is obtained through a constrained mountain pass and a constrained linking respectively.
63 citations
TL;DR: In this paper, the authors propose an extension of DMD that exploits low-rank tensor decompositions of potentially high-dimensional data sets to compute the corresponding DMD modes and eigenvalues.
Abstract: Dynamic mode decomposition (DMD) is a recently developed tool for the analysis of the behavior of complex dynamical systems. In this paper, we will propose an extension of DMD that exploits low-rank tensor decompositions of potentially high-dimensional data sets to compute the corresponding DMD modes and eigenvalues. The goal is to reduce the computational complexity and also the amount of memory required to store the data in order to mitigate the curse of dimensionality. The efficiency of these tensor-based methods will be illustrated with the aid of several different fluid dynamics problems such as the von Karman vortex street and the simulation of two merging vortices.
58 citations
TL;DR: In this article, the NLS Cauchy problem for periodic initial perturbations of the unstable background solution of NLS exciting just one of the instability modes is studied, in which only the corresponding unstable gap is theoretically open and the solution describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI.
Abstract: The focusing NLS equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, considered the main physical mechanism for the appearance of rogue (anomalous) waves (RWs) in Nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for periodic initial perturbations of the unstable background solution of NLS exciting just one of the unstable modes. We distinguish two cases. In the case in which only the corresponding unstable gap is theoretically open, the solution describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and the nonlinear RW stages are described by the 1-breather Akhmediev solution, whose parameters, different at each RW appearance, are always given in terms of the initial data through elementary functions. If the number of unstable modes is >1, this uniform in t dynamics is sensibly affected by perturbations due to numerics and/or real experiments, provoking O(1) corrections to the result. In the second case in which more than one unstable gap is open, a detailed investigation of all these gaps is necessary to get a uniform in $t$ dynamics, and this study is postponed to a subsequent paper. It is however possible to obtain the elementary description of the first nonlinear stage of MI, given again by the Akhmediev 1-breather solution, and how perturbations due to numerics and/or real experiments can affect this result.
56 citations
TL;DR: In this paper, the interplay of chemotaxis, convection and reaction terms is studied in the particular framework of a refined model for coral broadcast spawning, consisting of three equations describing the population densities of unfertilized sperms and eggs and the concentration of a chemical released by the latter, coupled to the incompressible Navier-Stokes equations.
Abstract: The interplay of chemotaxis, convection and reaction terms is studied in the particular framework of a refined model for coral broadcast spawning, consisting of three equations describing the population densities of unfertilized sperms and eggs and the concentration of a chemical released by the latter, coupled to the incompressible Navier–Stokes equations. Under mild assumptions on the initial data, global existence of classical solutions to an associated initial-boundary value problem in bounded planar domains is established. Moreover, all these solutions are shown to approach a spatially homogeneous equilibrium in the large time limit.
TL;DR: In this article, the stability properties of solitary wave solutions associated with Korteweg-de Vries-type models when the dispersion is very low were investigated using a compact, analytic approach and asymptotic perturbation theory.
Abstract: This paper sheds new light on the stability properties of solitary wave solutions associated with Korteweg–de Vries-type models when the dispersion is very low. Using a compact, analytic approach and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so a criterium of spectral instability of solitary waves is obtained for both models. Moreover, the nonlinear stability and spectral instability of the ground state solutions for both models is obtained for some specific regimen of parameters. Via a Lyapunov strategy and a variational analysis, we obtain the stability of the blow-up of solitary waves for the critical fractional KdV equation. The arguments presented in this investigation show promise for use in the study of the instability of traveling wave solutions of other nonlinear evolution equations.
TL;DR: In this article, the authors considered the parabolic system as a perturbation of a nonlocal parabolic equation and established a sufficient condition of the sensitivity function χ for the global existence of solutions under the assumption of smallness of the constant τ.
Abstract: This paper deals with time-global solutions to the parabolic system under the homogeneous Neumann boundary conditions in a bounded and convex domain () with smooth boundary . Here τ is a positive parameter, χ is a smooth function on satisfying and is a pair of nonnegative initial data. We will consider the above system as a perturbation of a nonlocal parabolic equation and establish a sufficient condition of the sensitivity function χ for the global existence of solutions under the assumption of smallness of the constant τ.
TL;DR: In this paper, the authors prove a fiberwise almost sure invariance principle for random piecewise expanding transformations in one and higher dimensions using recent developments on martingale techniques, which is based on the fiber-wise invariance theorem.
Abstract: We prove a fiberwise almost sure invariance principle for random piecewise expanding transformations in one and higher dimensions using recent developments on martingale techniques.
TL;DR: In this paper, the robustly transitive diffeomorphisms constructed by Bonatti and Viana have unique equilibrium states for natural classes of potentials and characterize the SRB measure as the unique equilibrium state for a suitable geometric potential.
Abstract: We show that the robustly transitive diffeomorphisms constructed by Bonatti and Viana have unique equilibrium states for natural classes of potentials. In particular, we characterize the SRB measure as the unique equilibrium state for a suitable geometric potential. The techniques developed are applicable to a wide class of DA diffeomorphisms, and persist under $C^1$ perturbations of the map. These results are an application of general machinery developed by the first and last named authors.
TL;DR: In this article, the authors considered a two-species chemotaxis system with two chemicals in a smooth bounded domain, subject to the non-flux boundary condition, and obtained a blowup criterion that if, then there exist finite time blow-up solutions to the system with and.
Abstract: This paper considers the two-species chemotaxis system with two chemicals in a smooth bounded domain , subject to the non-flux boundary condition, and . We obtain a blow-up criterion that if , then there exist finite time blow-up solutions to the system with and . When , the blow-up criterion becomes , and the global boundedness of solutions is furthermore established with under the condition that . This improves the current results for finite time blow-up with and global boundedness with respectively in Tao and Winkler (2015 Discrete Contin. Dyn. Syst. Ser. B 20 3165–83)
TL;DR: In this article, the authors considered a high-dimensional, chaotic climate model with positive Lyapunov exponents and showed that a boundary crisis in the Lorenz flow coincides with the approach to the unit circle of the eigenvalues of these operators associated with motions about the attractor, the stable resonances.
Abstract: The destruction of a chaotic attractor leading to rough changes in the dynamics of a dynamical system is studied. Local bifurcations are characterised by a single or a pair of characteristic exponents crossing the imaginary axis. The approach of such bifurcations in the presence of noise can be inferred from the slowing down of the correlation decay. On the other hand, little is known about global bifurcations involving high-dimensional attractors with positive Lyapunov exponents.
The global stability of chaotic attractors may be characterised by the spectral properties of the Koopman or the transfer operators governing the evolution of statistical ensembles. It has recently been shown that a boundary crisis in the Lorenz flow coincides with the approach to the unit circle of the eigenvalues of these operators associated with motions about the attractor, the stable resonances. A second type of resonances, the unstable resonances, is responsible for the decay of correlations and mixing on the attractor. In the deterministic case, those cannot be expected to be affected by general boundary crises.
Here, however, we give an example of chaotic system in which slowing down of the decay of correlations of some observables does occur at the approach of a boundary crisis. The system considered is a high-dimensional, chaotic climate model of physical relevance. Moreover, coarse-grained approximations of the transfer operators on a reduced space, constructed from a long time series of the system, give evidence that this behaviour is due to the approach of unstable resonances to the unit circle. That the unstable resonances are affected by the crisis can be physically understood from the fact that the process responsible for the instability, the ice-albedo feedback, is also active on the attractor. Implications regarding response theory and the design of early-warning signals are discussed.
TL;DR: In this article, non-cooperative parabolic reaction-diffusion systems with structural similarities with the scalar Fisher-KPP equation were studied and the existence of an extinction and persistence dichotomy was shown.
Abstract: This paper is concerned with non-cooperative parabolic reaction–diffusion systems which share structural similarities with the scalar Fisher–KPP equation. These similarities make it possible to prove, among other results, an extinction and persistence dichotomy and, when persistence occurs, the existence of a positive steady state, the existence of traveling waves with a half-line of possible speeds and a positive minimal speed and the equality between this minimal speed and the spreading speed for the Cauchy problem. Non-cooperative KPP systems can model various phenomena where the following three mechanisms occur: local diffusion in space, linear cooperation and su-perlinear competition.
TL;DR: In this paper, the basic reproduction number R 0 is defined and shown to be a threshold parameter: if R 0 > 1, the model has a unique globally stable positive equilibrium.
Abstract: In this paper, we study a reaction–diffusion vector-host epidemic model. We define the basic reproduction number R 0 and show that R 0 is a threshold parameter: if the disease free equilibrium is globally stable; if R 0 > 1 the model has a unique globally stable positive equilibrium. Our proof combines arguments from monotone dynamical system theory, persistence theory, and the theory of asymptotically autonomous semiflows.
TL;DR: In this article, it was shown that if the initial density decays not too slow at infinity, the 2D Cauchy problem of the density-dependent Navier-Stokes equations on the whole space admits a unique global strong solution.
Abstract: We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Navier–Stokes equations with vacuum as far-field density. It is proved that if the initial density decays not too slow at infinity, the 2D Cauchy problem of the density-dependent Navier–Stokes equations on the whole space admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Furthermore, we also obtain the large time decay rates of the spatial gradients of the velocity and the pressure, which are the same as those of the homogeneous case.
TL;DR: In this paper, the authors study a diffuse interface model of the two-component Hele-Shaw flow and prove that the corresponding problem is globally well posed with respect to a natural notion of weak solution.
Abstract: We study a diffuse interface model of the two-component Hele–Shaw flow. This is an advective Cahn–Hilliard equation for the relative concentration , where the incompressible velocity field is determined by the Darcy's law depending on the Korteweg force . Here μ is the derivative of a nonlocal non-convex free energy characterized by a logarithmic potential. The system is subject to no-flux boundary conditions for and μ along with an initial condition for . First of all, we prove that the corresponding problem is globally well posed with respect to a natural notion of weak solution. Also, we establish the existence of global strong solutions. In dimension two, we show the validity of the so-called instantaneous separation property. This means that any solution, which is not a pure phase initially, stays away from the pure phases, uniformly with respect to the initial energy and total mass. Finally we prove the existence of the global attractor for the corresponding dynamical system as well as the convergence to a single equilibrium of any weak solution in the two-dimensional case.
TL;DR: In this article, the authors present a general setting in which the formula describing the linear response of the physical measure of a perturbed system can be obtained and apply their results to expanding circle maps.
Abstract: We present a general setting in which the formula describing the linear response of the physical measure of a perturbed system can be obtained. In this general setting we obtain an algorithm to rigorously compute the linear
response. We apply our results to expanding circle maps. In particular, we present examples where we compute, up to a pre-specified error in the L1-norm, the response of expanding circle maps under stochastic and deterministic
perturbations. Moreover, we present an example where we compute, up to a
pre-specified error in the L1-norm, the response of the intermittent family at the boundary; i.e., when the unperturbed system is the doubling map.
TL;DR: In this article, the authors consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a non-uniform synchronisation and (III) an attractor without synchronisation.
Abstract: We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor without synchronisation of trajectories. The random attractors in phases (I) and (II) are random equilibrium points with negative Lyapunov exponents while in phase (III) there is a so-called random strange attractor with positive Lyapunov exponent. We analyse the occurrence of the different dynamical phases as a function of the linear stability of the origin (deterministic Hopf bifurcation parameter) and shear (amplitude-phase coupling parameter). We show that small shear implies synchronisation and obtain that synchronisation cannot be uniform in the absence of linear stability at the origin or in the presence of sufficiently strong shear. We provide numerical results in support of a conjecture that irrespective of the linear stability of the origin, there is a critical strength of the shear at which the system dynamics loses synchronisation and enters phase (III).
TL;DR: In this paper, a weighted version of the Caffarelli-Kohn-Nirenberg inequality in the framework of variable exponents is presented. And the combination of this inequality with a variant of the fountain theorem, yields the existence of infinitely many solutions for a class of non-homogeneous problems with Dirichlet boundary condition.
Abstract: We present a weighted version of the Caffarelli–Kohn–Nirenberg inequality in the framework of variable exponents. The combination of this inequality with a variant of the fountain theorem, yields the existence of infinitely many solutions for a class of non-homogeneous problems with Dirichlet boundary condition.
TL;DR: In this article, the first and second derivatives of the Lennard-Jones lattice energy per point in two dimensions were computed and sufficient conditions for the local minimality of these lattices were given.
Abstract: In this paper, we focus on finite Bravais lattice energies per point in two dimensions. We compute the first and second derivatives of these energies. We prove that the Hessian at the square and the triangular lattice are diagonal and we give simple sufficient conditions for the local minimality of these lattices. Furthermore, we apply our result to Lennard–Jones type interacting potentials that appear to be accurate in many physical and biological models. The goal of this investigation is to understand how the minimum of the Lennard–Jones lattice energy varies with respect to the density of the points. Considering the lattices of fixed area A, we find the maximal open set to which A must belong so that the triangular lattice is a minimizer (resp. a maximizer) among lattices of area A. Similarly, we find the maximal open set to which A must belong so that the square lattice is a minimizer (resp. a saddle point). Finally, we present a complete conjecture, based on numerical investigations and rigorous results among rhombic and rectangular lattices, for the minimality of the classical Lennard–Jones energy per point with respect to its area. In particular, we prove that the minimizer is a rectangular lattice if the area is sufficiently large.
TL;DR: Localized spatial patterns commonly occur for various classes of linear and nonlinear diffusive processes as mentioned in this paper, where the solution concentrates at discrete points in the domain, occur in the nonlinear reactiondiffusion (RD) modeling of diverse phenomena such as chemical patterns, biological morphogenesis, and the spatial distribution of urban crime.
Abstract: Localized spatial patterns commonly occur for various classes of linear and nonlinear diffusive processes. In particular, localized spot patterns, where the solution concentrates at discrete points in the domain, occur in the nonlinear reactiondiffusion (RD) modeling of diverse phenomena such as chemical patterns, biological morphogenesis, and the spatial distribution of urban crime. In a 2-D spatial domain we survey some recent and new results for the existence, linear stability, and slow dynamics of localized spot patterns by using the Brusselator RD model as the prototypical example. In the context of linear diffusive systems with localized solution behavior, we will discuss some previous results for the determination of the mean first capture time for a Brownian particle in a 2-D domain with localized traps, and the determination of the persistence threshold of a species in a 2-D landscape with patchy food resources. Common features in the analysis of all of these spatially localized patterns are emphasized, including the key role of certain matrices involving various Green’s functions, and the derivation and study of new classes of interacting particle systems and discrete variational problems arising from various asymptotic reductions. The mathematical tools include matched asymptotic analysis based on strong localized perturbation theory, spectral analysis, the analysis of nonlocal eigenvalue problems, and bifurcation theory. Some specific open problems are highlighted and, more broadly, we will discuss a few new research frontiers for the analysis of localized patterns in multi-dimensional domains.
TL;DR: In this article, the authors studied the nodal intersections number of random Gaussian toral Laplace eigenfunctions against a fixed smooth reference curve, and showed that the expected intersection number is proportional to the the square root of the eigenvalue times the length of curve, independent of its geometry.
Abstract: We study the nodal intersections number of random Gaussian toral Laplace eigenfunctions ("arithmetic random waves") against a fixed smooth reference curve. The expected intersection number is proportional to the the square root of the eigenvalue times the length of curve, independent of its geometry. The asymptotic behaviour of the variance was addressed by Rudnick-Wigman; they found a precise asymptotic law for "generic" curves with nowhere vanishing curvature, depending on both its geometry and the angular distribution of lattice points lying on circles corresponding to the Laplace eigenvalue. They also discovered that there exist peculiar "static" curves, with variance of smaller order of magnitude, though did not prescribe what the true asymptotic behaviour is in this case. In this paper we study the finer aspects of the limit distribution of the nodal intersections number. For "generic" curves we prove the Central Limit Theorem (at least, for "most" of the energies). For the aforementioned static curves we establish a non-Gaussian limit theorem for the distribution of nodal intersections, and on the way find the true asymptotic behaviour of their fluctuations, under the well-separatedness assumption on the corresponding lattice points, satisfied by most of the eigenvalues.
TL;DR: In this article, the authors considered the parabolic-parabolic Patlak-Keller-Segel model with advection by a large strictly monotone shear flow.
Abstract: In this paper we consider the parabolic–parabolic Patlak–Keller–Segel models in with advection by a large strictly monotone shear flow. Without the shear flow, the model is L 1 critical in two dimensions with critical mass : solutions with mass less than are global in time and there exist solutions with mass larger than which blow up in finite time (Schweyer 2014 (arXiv:1403.4975)). We show that the additional shear flow, if it is chosen sufficiently large, suppresses one dimension of the dynamics and hence can suppress blow-up. In contrast with the parabolic-elliptic case (Bedrossian and He 2016 SIAM J. Math. Anal. 49 4722–66), the strong shear flow has destabilizing effect in addition to the enhanced dissipation effect, which makes the problem more difficult.
TL;DR: In this article, the authors study the Riemann problem for multidimensional compressible isentropic Euler equations using the framework developed in Chiodaroli et al. using the techniques of De Lellis and Szekelyhidi.
Abstract: We study the Riemann problem for multidimensional compressible isentropic Euler equations Using the framework developed in Chiodaroli et al (2015 Commun Pure Appl Math 68 1157–90), and based on the techniques of De Lellis and Szekelyhidi (2010 Arch Ration Mech Anal 195 225–60), we extend the results of Chiodaroli and Kreml (2014 Arch Ration Mech Anal 214 1019–49) and prove that it is possible to characterize a set of Riemann data, giving rise to a self-similar solution consisting of one admissible shock and one rarefaction wave, for which the problem also admits infinitely many admissible weak solutions
TL;DR: In this paper, a shadow-type limit of a reaction-diffusion-ODE model was used to show that the instability driven by nonlocal terms may lead to the formation of unbounded spike patterns.
Abstract: Coupling a reaction-diffusion equation with ordinary differential equa- tions (ODE) may lead to diffusion-driven instability (DDI) which, in contrast to the classical reaction-diffusion models, causes destabilization of both, constant solutions and Turing patterns. Using a shadow-type limit of a reaction-diffusion-ODE model, we show that in such cases the instability driven by nonlocal terms (a counterpart of DDI) may lead to formation of unbounded spike patterns.