Pub Date : 2023-09-18DOI: 10.1017/s0956792523000256
Luís Almeida, Pierre-Alexandre Bliman, Nga Nguyen, Nicolas Vauchelet
Abstract In this paper, we investigate an initial-boundary value problem of a reaction–diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase plane analysis, the present paper studies the existence and properties of non-constant steady-state solutions depending on several parameters. Then, we prove some sufficient conditions for their stability. We show that the long-time efficiency of this control method depends strongly on the size of the treated zone and the migration rate. To illustrate these theoretical results, we provide some numerical simulations in the framework of mosquito population control.
{"title":"Steady-state solutions for a reaction–diffusion equation with Robin boundary conditions: Application to the control of dengue vectors","authors":"Luís Almeida, Pierre-Alexandre Bliman, Nga Nguyen, Nicolas Vauchelet","doi":"10.1017/s0956792523000256","DOIUrl":"https://doi.org/10.1017/s0956792523000256","url":null,"abstract":"Abstract In this paper, we investigate an initial-boundary value problem of a reaction–diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase plane analysis, the present paper studies the existence and properties of non-constant steady-state solutions depending on several parameters. Then, we prove some sufficient conditions for their stability. We show that the long-time efficiency of this control method depends strongly on the size of the treated zone and the migration rate. To illustrate these theoretical results, we provide some numerical simulations in the framework of mosquito population control.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135109331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-13DOI: 10.1017/s0956792523000165
Stephen Anco, Philip Broadbridge
{"title":"Editorial “Symmetries and differential equations”","authors":"Stephen Anco, Philip Broadbridge","doi":"10.1017/s0956792523000165","DOIUrl":"https://doi.org/10.1017/s0956792523000165","url":null,"abstract":"","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134990982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-06DOI: 10.1017/s0956792523000244
M. G. Hennessy, Giulia L. Celora, Sarah L. Waters, A. Münch, B. Wagner
Mathematical models of polyelectrolyte gels are often simplified by assuming the gel is electrically neutral. The rationale behind this assumption is that the thickness of the electric double layer (EDL) at the free surface of the gel is small compared to the size of the gel. Hence, the thin-EDL limit is taken, in which the thickness of the EDL is set to zero. Despite the widespread use of the thin-EDL limit, the solutions in the EDL are rarely computed and shown to match to the solutions for the electrically neutral bulk. The aims of this paper are to study the structure of the EDL and establish the validity of the thin-EDL limit. The model for the gel accounts for phase separation, which gives rise to diffuse interfaces with a thickness described by the Kuhn length. We show that the solutions in the EDL can only be asymptotically matched to the solutions for an electrically neutral bulk, in general, when the Debye length is much smaller than the Kuhn length. If the Debye length is similar to or larger than the Kuhn length, then phase separation can be initiated in the EDL. This phase separation spreads into the bulk of the gel and gives rise to electrically charged layers with different degrees of swelling. Thus, the thin-EDL limit and the assumption of electroneutrality only generally apply when the Debye length is much smaller than the Kuhn length.
{"title":"Breakdown of electroneutrality in polyelectrolyte gels","authors":"M. G. Hennessy, Giulia L. Celora, Sarah L. Waters, A. Münch, B. Wagner","doi":"10.1017/s0956792523000244","DOIUrl":"https://doi.org/10.1017/s0956792523000244","url":null,"abstract":"\u0000 Mathematical models of polyelectrolyte gels are often simplified by assuming the gel is electrically neutral. The rationale behind this assumption is that the thickness of the electric double layer (EDL) at the free surface of the gel is small compared to the size of the gel. Hence, the thin-EDL limit is taken, in which the thickness of the EDL is set to zero. Despite the widespread use of the thin-EDL limit, the solutions in the EDL are rarely computed and shown to match to the solutions for the electrically neutral bulk. The aims of this paper are to study the structure of the EDL and establish the validity of the thin-EDL limit. The model for the gel accounts for phase separation, which gives rise to diffuse interfaces with a thickness described by the Kuhn length. We show that the solutions in the EDL can only be asymptotically matched to the solutions for an electrically neutral bulk, in general, when the Debye length is much smaller than the Kuhn length. If the Debye length is similar to or larger than the Kuhn length, then phase separation can be initiated in the EDL. This phase separation spreads into the bulk of the gel and gives rise to electrically charged layers with different degrees of swelling. Thus, the thin-EDL limit and the assumption of electroneutrality only generally apply when the Debye length is much smaller than the Kuhn length.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43129757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-31DOI: 10.1017/s0956792523000232
B. Carbonaro, Marco Menale
The present paper deals with the kinetic-theoretic description of the evolution of systems consisting of many particles interacting not only with each other but also with the external world, so that the equation governing their evolution contains an additional term representing such interaction, called the ‘forcing term’. Firstly, the interactions between pairs of particles are both conservative and nonconservative; the latter represents, among others, birth/death rates. The ‘forcing term’ does not express a ‘classical’ force exerted by the external world on the particles, but a more general influence on the effects of mutual interactions of particles, for instance, climate changes, that increase or decrease the different agricultural productions at different times, thus altering the economic relationships between different subsystems, that in turn can be also perturbed by stock market fluctuations, sudden wars, periodic epidemics, and so on. Thus, the interest towards these problems moves the mathematical analysis of the effects of different kinds of forcing terms on solutions to equations governing the collective (that is statistical) behaviour of such nonconservative many-particle systems. In the present paper, we offer a study of the basic mathematical properties of such solutions, along with some numerical simulations to show the effects of forcing terms for a classical prey–predator model in ecology.
{"title":"A nonconservative kinetic framework under the action of an external force field: Theoretical results with application inspired to ecology","authors":"B. Carbonaro, Marco Menale","doi":"10.1017/s0956792523000232","DOIUrl":"https://doi.org/10.1017/s0956792523000232","url":null,"abstract":"\u0000 The present paper deals with the kinetic-theoretic description of the evolution of systems consisting of many particles interacting not only with each other but also with the external world, so that the equation governing their evolution contains an additional term representing such interaction, called the ‘forcing term’. Firstly, the interactions between pairs of particles are both conservative and nonconservative; the latter represents, among others, birth/death rates. The ‘forcing term’ does not express a ‘classical’ force exerted by the external world on the particles, but a more general influence on the effects of mutual interactions of particles, for instance, climate changes, that increase or decrease the different agricultural productions at different times, thus altering the economic relationships between different subsystems, that in turn can be also perturbed by stock market fluctuations, sudden wars, periodic epidemics, and so on. Thus, the interest towards these problems moves the mathematical analysis of the effects of different kinds of forcing terms on solutions to equations governing the collective (that is statistical) behaviour of such nonconservative many-particle systems. In the present paper, we offer a study of the basic mathematical properties of such solutions, along with some numerical simulations to show the effects of forcing terms for a classical prey–predator model in ecology.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47466315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-11DOI: 10.1017/s0956792523000220
Qiaoling Chen, Sanyi Tang, Z. Teng, Feng Wang
A delayed reaction-diffusion system with free boundaries is investigated in this paper to understand how the bacteria spread spatially to larger area from the initial infected habitat. Under the assumptions that the nonlinearities are of monostable type and the initial values satisfy some compatible condition, we show that the free boundary problem is well-posed and discuss the long-time behaviour of solution (including spreading and vanishing) in terms of the spatial-temporal risk index. Furthermore, to determine the spreading speed of free boundaries when spreading occurs, we first study the distribution of roots of a transcendental equation containing a polynomial of degree four and then establish the existence and uniqueness of monotone solution to a delay-induced nonlocal semi-wave problem by employing the approximation method, lower-upper solutions technique and Schauder fixed point theorem. It is shown that time delays slow down the spreading of bacteria.
{"title":"Spreading dynamics of a diffusive epidemic model with free boundaries and two delays","authors":"Qiaoling Chen, Sanyi Tang, Z. Teng, Feng Wang","doi":"10.1017/s0956792523000220","DOIUrl":"https://doi.org/10.1017/s0956792523000220","url":null,"abstract":"\u0000 A delayed reaction-diffusion system with free boundaries is investigated in this paper to understand how the bacteria spread spatially to larger area from the initial infected habitat. Under the assumptions that the nonlinearities are of monostable type and the initial values satisfy some compatible condition, we show that the free boundary problem is well-posed and discuss the long-time behaviour of solution (including spreading and vanishing) in terms of the spatial-temporal risk index. Furthermore, to determine the spreading speed of free boundaries when spreading occurs, we first study the distribution of roots of a transcendental equation containing a polynomial of degree four and then establish the existence and uniqueness of monotone solution to a delay-induced nonlocal semi-wave problem by employing the approximation method, lower-upper solutions technique and Schauder fixed point theorem. It is shown that time delays slow down the spreading of bacteria.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49497263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-08DOI: 10.1017/s0956792523000219
E. Luckins, C. Breward, I. Griffiths, C. Please
A porous material that has been contaminated with a hazardous chemical agent is typically decontaminated by applying a cleanser solution to the surface and allowing the cleanser to react into the porous material, neutralising the agent. The agent and cleanser are often immiscible fluids and so, if the porous material is initially saturated with agent, a reaction front develops with the decontamination reaction occurring at this interface between the fluids. We investigate the effect of different initial agent configurations within the pore space on the decontamination process. Specifically, we compare the decontamination of a material initially saturated by the agent with the situation when, initially, the agent only coats the walls of the pores (referred to as the ‘agent-on-walls’ case). In previous work (Luckins et al., European Journal of Applied Mathematics, 31(5):782–805, 2020), we derived homogenised models for both of these decontamination scenarios, and in this paper we explore the solutions of these two models. We find that, for an identical initial volume of agent, the decontamination time is generally much faster for the agent-on-walls case compared with the initially saturated case, since the surface area on which the reaction can occur is greater. However for sufficiently deep spills of contaminant, or sufficiently slow reaction rates, decontamination in the agent-on-walls scenario can be slower. We also show that, in the limit of a dilute cleanser with a deep initial agent spill, the agent-on-walls model exhibits behaviour akin to a Stefan problem of the same form as that arising in the initially saturated model. The decontamination time is shown to decrease with both the applied cleanser concentration and the rate of the chemical reaction. However, increasing the cleanser concentration is also shown to result in lower decontamination efficiency, with an increase in the amount of cleanser chemical that is wasted.
{"title":"The effect of pore-scale contaminant distribution on the reactive decontamination of porous media","authors":"E. Luckins, C. Breward, I. Griffiths, C. Please","doi":"10.1017/s0956792523000219","DOIUrl":"https://doi.org/10.1017/s0956792523000219","url":null,"abstract":"\u0000 A porous material that has been contaminated with a hazardous chemical agent is typically decontaminated by applying a cleanser solution to the surface and allowing the cleanser to react into the porous material, neutralising the agent. The agent and cleanser are often immiscible fluids and so, if the porous material is initially saturated with agent, a reaction front develops with the decontamination reaction occurring at this interface between the fluids. We investigate the effect of different initial agent configurations within the pore space on the decontamination process. Specifically, we compare the decontamination of a material initially saturated by the agent with the situation when, initially, the agent only coats the walls of the pores (referred to as the ‘agent-on-walls’ case). In previous work (Luckins et al., European Journal of Applied Mathematics, 31(5):782–805, 2020), we derived homogenised models for both of these decontamination scenarios, and in this paper we explore the solutions of these two models. We find that, for an identical initial volume of agent, the decontamination time is generally much faster for the agent-on-walls case compared with the initially saturated case, since the surface area on which the reaction can occur is greater. However for sufficiently deep spills of contaminant, or sufficiently slow reaction rates, decontamination in the agent-on-walls scenario can be slower. We also show that, in the limit of a dilute cleanser with a deep initial agent spill, the agent-on-walls model exhibits behaviour akin to a Stefan problem of the same form as that arising in the initially saturated model. The decontamination time is shown to decrease with both the applied cleanser concentration and the rate of the chemical reaction. However, increasing the cleanser concentration is also shown to result in lower decontamination efficiency, with an increase in the amount of cleanser chemical that is wasted.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46549282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-07DOI: 10.1017/s0956792523000189
M. Stojanović, F. Romanò, H. Kuhlmann
In numerical linear stability investigations, the rates of change of the kinetic and thermal energy of the perturbation flow are often used to identify the dominant mechanisms by which kinetic or thermal energy is exchanged between the basic and the perturbation flow. Extending the conventional energy analysis for a single-phase Boussinesq fluid, the energy budgets of arbitrary infinitesimal perturbations to the basic two-phase liquid–gas flow are derived for an axisymmetric thermocapillary bridge when the material parameters in both phases depend on the temperature. This allows identifying individual transport terms and assessing their contributions to the instability if the basic flow and the critical mode are evaluated at criticality. The full closed-form energy budgets of linear modes have been derived for thermocapillary two-phase flow taking into account the temperature dependence of all thermophysical parameters. The influence of different approximations to the temperature dependence on the linear stability boundary of the axisymmetric flow in thermocapillary liquid bridges is tested regarding their accuracy. The general mechanism of symmetry breaking turns out to be very robust.
{"title":"Instability of axisymmetric flow in thermocapillary liquid bridges: Kinetic and thermal energy budgets for two-phase flow with temperature-dependent material properties","authors":"M. Stojanović, F. Romanò, H. Kuhlmann","doi":"10.1017/s0956792523000189","DOIUrl":"https://doi.org/10.1017/s0956792523000189","url":null,"abstract":"\u0000 In numerical linear stability investigations, the rates of change of the kinetic and thermal energy of the perturbation flow are often used to identify the dominant mechanisms by which kinetic or thermal energy is exchanged between the basic and the perturbation flow. Extending the conventional energy analysis for a single-phase Boussinesq fluid, the energy budgets of arbitrary infinitesimal perturbations to the basic two-phase liquid–gas flow are derived for an axisymmetric thermocapillary bridge when the material parameters in both phases depend on the temperature. This allows identifying individual transport terms and assessing their contributions to the instability if the basic flow and the critical mode are evaluated at criticality. The full closed-form energy budgets of linear modes have been derived for thermocapillary two-phase flow taking into account the temperature dependence of all thermophysical parameters. The influence of different approximations to the temperature dependence on the linear stability boundary of the axisymmetric flow in thermocapillary liquid bridges is tested regarding their accuracy. The general mechanism of symmetry breaking turns out to be very robust.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45520194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-06DOI: 10.1017/s0956792523000190
Hana Krakovská, C. Kuehn, Iacopo P Longo
{"title":"Resilience of dynamical systems – ERRATUM","authors":"Hana Krakovská, C. Kuehn, Iacopo P Longo","doi":"10.1017/s0956792523000190","DOIUrl":"https://doi.org/10.1017/s0956792523000190","url":null,"abstract":"","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":"77 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139362306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-14DOI: 10.1017/S0956792523000098
J. R. King, G. Richardson, J. Foster
Abstract The dynamics of interfaces in slow diffusion equations with strong absorption are studied. Asymptotic methods are used to give descriptions of the behaviour local to a comprehensive range of possible singular events that can occur in any evolution. These events are: when an interface changes its direction of propagation (reversing and anti-reversing), when an interface detaches from an absorbing obstacle (detaching), when two interfaces are formed by film rupture (touchdown) and when the solution undergoes extinction. Our account of extinction and self-similar reversing and anti-reversing is built upon previous work; results on non-self-similar reversing and anti-reversing and on the various types of detachment and touchdown are developed from scratch. In all cases, verification of the asymptotic results against numerical solutions to the full PDE is provided. Self-similar solutions, both of the full equation and of its asymptotic limits, play a central role in the analysis.
{"title":"Interface behaviour of the slow diffusion equation with strong absorption: Intermediate-asymptotic properties","authors":"J. R. King, G. Richardson, J. Foster","doi":"10.1017/S0956792523000098","DOIUrl":"https://doi.org/10.1017/S0956792523000098","url":null,"abstract":"Abstract The dynamics of interfaces in slow diffusion equations with strong absorption are studied. Asymptotic methods are used to give descriptions of the behaviour local to a comprehensive range of possible singular events that can occur in any evolution. These events are: when an interface changes its direction of propagation (reversing and anti-reversing), when an interface detaches from an absorbing obstacle (detaching), when two interfaces are formed by film rupture (touchdown) and when the solution undergoes extinction. Our account of extinction and self-similar reversing and anti-reversing is built upon previous work; results on non-self-similar reversing and anti-reversing and on the various types of detachment and touchdown are developed from scratch. In all cases, verification of the asymptotic results against numerical solutions to the full PDE is provided. Self-similar solutions, both of the full equation and of its asymptotic limits, play a central role in the analysis.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":"34 1","pages":"1099 - 1132"},"PeriodicalIF":1.9,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47960207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-22DOI: 10.1017/s0956792523000086
José R. Fernández, R. Quintanilla
In this paper, we consider the time decay of the solutions to some problems arising in strain gradient thermoelasticity. We restrict to the two-dimensional case, and we assume that two dissipative mechanisms are introduced, the temperature and the mass dissipation. First, we show that this problem is well-posed proving that the operator defining it generates a contractive semigroup of linear operators. Then, assuming that the function involving the coupling terms is elliptic, the exponential decay of the solutions is concluded as well as the analyticity of the solutions. Finally, we describe how to obtain the exponential stability in the case of hyperbolic dissipation.
{"title":"-coupling mechanisms are sufficient to obtain exponential decay in strain gradient elasticity","authors":"José R. Fernández, R. Quintanilla","doi":"10.1017/s0956792523000086","DOIUrl":"https://doi.org/10.1017/s0956792523000086","url":null,"abstract":"\u0000 In this paper, we consider the time decay of the solutions to some problems arising in strain gradient thermoelasticity. We restrict to the two-dimensional case, and we assume that two dissipative mechanisms are introduced, the temperature and the mass dissipation. First, we show that this problem is well-posed proving that the operator defining it generates a contractive semigroup of linear operators. Then, assuming that the function involving the coupling terms is elliptic, the exponential decay of the solutions is concluded as well as the analyticity of the solutions. Finally, we describe how to obtain the exponential stability in the case of hyperbolic dissipation.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48721117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：