Search results for "hyperbolic equation"

showing 4 items of 4 documents

Homogenization of the wave equation in composites with imperfect interface : a memory effect

2007

Abstract In this paper we study the asymptotic behaviour of the wave equation with rapidly oscillating coefficients in a two-component composite with e-periodic imperfect inclusions. We prescribe on the interface between the two components a jump of the solution proportional to the conormal derivatives through a function of order e γ . For the different values of γ, we obtain different limit problems. In particular, for γ = 1 we have a linear memory effect in the homogenized problem.

Mathematics(all)HomogenizationHomogenization; Hyperbolic equationsApplied MathematicsGeneral MathematicsHyperbolic equations010102 general mathematicsMathematical analysisComposite numberGeometryWave equation01 natural sciencesHomogenization (chemistry)periodic homogenization; wave equation; interface problem010101 applied mathematicsJump[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]wave equationinterface problemImperfect0101 mathematics[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]periodic homogenizationComputingMilieux_MISCELLANEOUSMathematics
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A PDE model for the spatial dynamics of a voles population structured in age

2020

Abstract We prove existence and stability of entropy weak solutions for a macroscopic PDE model for the spatial dynamics of a population of voles structured in age. The model consists of a scalar PDE depending on time, t , age, a , and space x = ( x 1 , x 2 ) , supplemented with a non-local boundary condition at a = 0 . The flux is linear with constant coefficient in the age direction but contains a non-local term in the space directions. Also, the equation contains a term of second order in the space variables only. Existence of solutions is established by compensated compactness, see Panov (2009), and we prove stability by a doubling of variables type argument.

Parabolic–hyperbolic equationEnergy estimateseducation.field_of_studyConstant coefficientsDoubling of variablesPopulation dynamics structured in age and spaceApplied Mathematics010102 general mathematicsPopulationMathematical analysis01 natural sciences010101 applied mathematicsCompact spaceNon-local fluxCompensated compactnessPopulation dynamics structured in age and space Parabolic–hyperbolic equation Non-local flux Boundary value problem Energy estimates Compensated compactness Doubling of variablesBoundary value problem0101 mathematicseducationBoundary value problemAnalysisMathematics
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AN HYPERBOLIC-PARABOLIC PREDATOR-PREY MODEL INVOLVING A VOLE POPULATION STRUCTURED IN AGE

2020

Abstract We prove existence and stability of entropy solutions for a predator-prey system consisting of an hyperbolic equation for predators and a parabolic-hyperbolic equation for preys. The preys' equation, which represents the evolution of a population of voles as in [2] , depends on time, t, age, a, and on a 2-dimensional space variable x, and it is supplemented by a nonlocal boundary condition at a = 0 . The drift term in the predators' equation depends nonlocally on the density of preys and the two equations are also coupled via classical source terms of Lotka-Volterra type, as in [4] . We establish existence of solutions by applying the vanishing viscosity method, and we prove stabil…

Population dynamicsPopulationType (model theory)Space (mathematics)01 natural sciencesStability (probability)Predator-prey systemsNonlinear Sciences::Adaptation and Self-Organizing SystemsApplied mathematicsQuantitative Biology::Populations and Evolution[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]0101 mathematicseducationEntropy (arrow of time)Variable (mathematics)Mathematicseducation.field_of_studyApplied Mathematics010102 general mathematicsNonlocal boundary value problemNonlocal conservation lawsParabolic-hyperbolic equationsTerm (time)010101 applied mathematicsPopulation dynamics Predator-prey systems Parabolic-hyperbolic equations Nonlocal conservation laws Nonlocal boundary value problemHyperbolic partial differential equationAnalysis
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On the Cauchy problem for microlocally symmetrizable hyperbolic systems with log-Lipschitz coefficients

2017

International audience; The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in space problem we establish energy estimates with finite loss of derivatives, which is linearly increasing in time. This implies well-posedness in H ∞ , if the coefficients enjoy enough smoothness in x. From this result, by standard arguments (i.e. extension and convexification) we deduce also local existence and uniqueness. A huge part of the analysis is devoted to give an appropriate sense to the Cauchy problem, which is not evide…

Pure mathematicsloss of derivativeshyperbolic equationGeneral MathematicsMathematics::Analysis of PDEsmicrolocal symmetrizabilityhyperbolic equations; hyperbolic systems; log-lipschitz coefficientsSpace (mathematics)01 natural sciencesMathematics - Analysis of PDEslog-Lipschitz regularity; loss of derivatives; global and local Cauchy problem; well-posedness; non-characteristic Cauchy problemwell-posednessFOS: MathematicsInitial value problem[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]Uniqueness0101 mathematics[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]MathematicsSmoothness (probability theory)Spacetimelog-lipschitz coefficients010102 general mathematicsglobal and local Cauchy problemExtension (predicate logic)Lipschitz continuitynon-characteristic Cauchy problemhyperbolic equationshyperbolic systemMathematics Subject Classificationlog-Lipschitz regularityhyperbolic systemsAnalysis of PDEs (math.AP)
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