Search results for "energy estimates"

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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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Decay estimates for time-fractional and other non-local in time subdiffusion equations in R^d

2016

We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in R d . An important special case is the timefractional diffusion equation, which has seen much interest during the last years, mostly due to its applications in the modeling of anomalous diffusion processes. We follow three different approaches and techniques to study this particular case: (A) estimates based on the fundamental solution and Young’s inequality, (B) Fourier multiplier methods, and (C) the energy method. It turns out that the decay behaviour is markedly different from the heat equation case, in particular there occurs a critical dimension phenom…

fundamental solutionFourier multipliersubdiffusionenergy estimatestime-fractional diffusionultraslow diffusionsubordinationtemporal decay estimates
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