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Register a new userPhysiologically structured populations with diffusion and dynamic boundary conditions
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We consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. The model is equipped with generalized Wentzell-Robin (or dynamic) boundary conditions. This allows modelling of adhesion at extremely small or large sizes. We establish existence and positivity of solutions by showing that solutions are governed by a positive quasicontractive semigroup of linear operators on the biologically relevant state space. This is carried out via establishing dissipativity of a suitably perturbed semigroup generator. We also show that solutions of the model exhibit balanced exponential growth, that is our model admits a finite dimensional global attractor. In case of strictly positive fertility we are able to establish that solutions in fact exhibit asynchronous exponential growth.
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In this paper we develop an existence theory for the nonlinear initial-boundary value problem with singular diffusion $partial_t u = text{div}(k(x) abla G(u))$, $u|_{t=0}=u_0$ with Neumann boundary conditions $k(x) abla G(u)cdot u = 0$. Here $xin Bsubset mathbb{R}^d$, a bounded open set with locally Lipchitz boundary, and with $ u$ as the unit outer normal. The function $G$ is Lipschitz continuous and nondecreasing, while $k(x)$ is diagonal matrix. We show that any two weak entropy solutions $u$ and $v$ satisfy $Vert{u(t)-v(t)}Vert_{L^1(B)}le Vert{u|_{t=0}-v|_{t=0}}Vert_{L^1(B)}e^{Ct}$, for almost every $tge 0$, and a constant $C=C(k,G,B)$. If we restrict to the case when the entries $k_i$ of $k$ depend only on the corresponding component, $k_i=k_i(x_i)$, we show that there exists an entropy solution, thus establishing in this case that the problem is well-posed in the sense of Hadamard.
Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral theory of semigroups to establish conditions for the existence of a positive equilibrium solution of our model. Then, we formulate conditions that guarantee that the linearised system is governed by a positive quasicontraction semigroup on the biologically relevant state space. We also show that the governing linear semigroup is eventually compact, hence growth properties of the semigroup are determined by the spectrum of its generator. In the case of a separable fertility function, we deduce a characteristic equation, and investigate the stability of equilibrium solutions in the general case using positive perturbation arguments.
This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.
In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general boundary condition of dynamic type. This type of condition prescribes some kind of mass conservation; hence extinction effects are not expected for solutions that emanate from strictly positive initial data. Our main results regard existence of weak solutions, instantaneous regularization properties, long-time behavior, and, under special conditions, uniqueness.
We consider parabolic systems with nonlinear dynamic boundary conditions, for which we give a rigorous derivation. Then, we give them several physical interpretations which includes an interpretation for the porous-medium equation, and for certain reaction-diffusion systems that occur in mathematical biology and ecology. We devise several strategies which imply (uniform)}$L^{p} and}$L^{infty}$ estimates on the solutions for the initial value problems considered.
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