Parabolic-elliptic System Research Articles

Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model

A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other is not. A complete Lie symmetry classification, including a number of the cases characterised as being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications (tumour growth). Graphical representations of the solutions are provided and a biological interpretation is briefly addressed. The results are generalised on multi-dimensional case under the assumption of the radially symmetrical shape of the tumour.

Weak solutions for a thermoelectric problem with power-type boundary effects

This paper deals with thermoelectric problems including the Peltier and Seebeck effects. The coupled elliptic and doubly quasilinear parabolic equations for the electric and heat currents are stated, respectively, accomplished with power-type boundary conditions that describe the thermal radiative effects. To verify the existence of weak solutions to this coupled problem (Theorem 1), analytical investigations for abstract multi-quasilinear elliptic-parabolic systems with nonsmooth data are presented (Theorem 2 and 3). They are essentially approximated solutions based on the Rothe method. It consists on introducing time discretized problems, establishing their existence, and then passing to the limit as the time step goes to zero. The proof of the existence of time discretized solutions relies on fixed point and compactness arguments. In this study, we establish quantitative estimates to clarify the smallness conditions.

On the entropy method and exponential convergence to equilibrium for a recombination–drift–diffusion system with self-consistent potential

We consider a Shockley–Read–Hall recombination–drift–diffusion model coupled to Poisson’s equation and subject to boundary conditions, which imply conservation of the total charge. As main result, we derive an explicit functional inequality between relative entropy and entropy production rate, which implies exponential convergence to equilibrium with explicit constant and rate. We report that the key entropy–entropy production inequality ought rather not to be formulated on the states space of the parabolic–elliptic system, but on the reduced states space of the associated nonlocal drift–diffusion problem, where the Poisson equation is replaced by the corresponding Green function.

Inter-species competition and chemorepulsion

We study an extension of the Lotka–Volterra competition model in which one of the competing species avoids encounters with rivals by means of chemorepulsion—a repulsive reaction to the scent of competitors. The evolution of the species densities is described in terms of parabolic equations with cross-diffusive and competitive terms. The chemical signal used to detect rivals may diffuse in the environment leading to a fully parabolic problem or to a parabolic–elliptic system if the diffusion of the chemical is dominant. The most difficult case from the viewpoint of well-posedness and global existence is the parabolic–ODE system which corresponds to the case when the chemical signal does not diffuse in the environment. We prove global existence of solutions in the three cases for any space dimension for a wide range of parameters and initial conditions. For the parabolic–elliptic case, the moving rectangles method is adapted to prove the converge to a constant steady state for some range of parameters and initial data. Linear stability analysis indicates that the constant steady state in the fully parabolic case loses stability when the strength of chemorepulsion is too high and that sufficiently low degradation rate of chemorepellent stabilizes the constant steady state.

Blow-up for the solutions to nonlinear parabolic–elliptic system modeling chemotaxis

In this paper, we study the blow-up solutions for the simplified Keller–Segel system modeling chemotaxis. We prove that there is the occurrence of [Formula: see text] blow-up to the radially symmetric solutions. We also prove that blow-up occur only at the point [Formula: see text].

Well-posedness and inverse Robin estimate for a multiscale elliptic/parabolic system

We establish the well-posedness of a coupled micro–macro parabolic–elliptic system modeling the interplay between two pressures in a gas–liquid mixture close to equilibrium that is filling a porous media with distributed microstructures. Additionally, we prove a local stability estimate for the inverse micro–macro Robin problem, potentially useful in identifying quantitatively a micro–macro interfacial Robin transfer coefficient given microscopic measurements on accessible fixed interfaces. To tackle the solvability issue we use two-scale energy estimates and two-scale regularity/compactness arguments cast in the Schauder’s fixed point theorem. A number of auxiliary problems, regularity, and scaling arguments are used in ensuring the suitable Fréchet differentiability of the solution and the structure of the inverse stability estimate.

A degenerate elliptic-parabolic system arising in competitive contaminant transport

The objective of this work is to study a coupled system of degenerate and nonlinear partial differential equations governing the transport of reactive solutes in groundwater. We show that this system admits a unique weak solution provided the nonlinear adsorption isotherm associated with the reaction process satisfies certain physically reasonable structural conditions, by addressing a more general problem. In addition, we conclude, that the solute concentrations stay non-negative if the source term is componentwise non-negative and investigate numerically the finite speed of propagation of compactly supported initial concentrations, in a two-component test case.

Boundedness of Solutions to a Parabolic-Elliptic Keller–Segel Equation in ℝ2 with Critical Mass

Abstract We consider the Cauchy problem for a parabolic-elliptic system in ℝ 2 {\mathbb{R}^{2}} , called the parabolic-elliptic Keller–Segel equation, which appears in various fields in biology and physics. In the critical mass case where the total mass of the initial data is 8 ⁢ π {8\pi} , the unboundedness of nonnegative solutions to the Cauchy problem was shown by Blanchet, Carrillo and Masmoudi [7] under some conditions on the initial data, on the other hand, conditions for boundedness were given by Blanchet, Carlen and Carrillo [6] and López-Gómez, Nagai and Yamada [23]. In this paper, we investigate further the boundedness of nonnegative solutions.

The variational formulation of the fully parabolic Keller–Segel system with degenerate diffusion

We prove the time-global existence of solutions of the degenerate Keller–Segel system in higher dimensions, under the assumption that the mass of the first component is below a certain critical value. What we deal with is the full parabolic–parabolic system rather than the simplified parabolic–elliptic system. Our approach is to formulate the problem as a gradient flow on the Wasserstein space. We first consider a time-discretized problem, in which the values of the solution are determined iteratively by solving a certain minimizing problem at each time step. Here we use a new minimizing scheme at each time level, which gives the time-discretized solutions favorable regularity properties. As a consequence, it becomes relatively easy to prove that the time-discretized solutions converge to a weak solution of the original system as the time step size tends to zero.

Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension

This article is devoted to the discussion of the boundary layer which arises from the one-dimensional parabolic elliptic type Keller-Segel system to the corresponding aggregation system on the half space case. The characteristic boundary layer is shown to be stable for small diffusion coefficients by using asymptotic analysis and detailed error estimates between the Keller-Segel solution and the approximate solution. In the end, numerical simulations for this boundary layer problem are provided.

Uniform in time interacting particle approximations for nonlinear equations of Patlak-Keller-Segel type

We study a system of interacting diffusions that models chemotaxis of biological cells or microorganisms (referred to as particles) in a chemical field that is dynamically modified through the collective contributions from the particles. Such systems of reinforced diffusions have been widely studied and their hydrodynamic limits that are nonlinear non-local partial differential equations are usually referred to as Patlak-Keller-Segel (PKS) equations. Solutions of the classical PKS equation may blow up in finite time and much of the PDE literature has been focused on understanding this blow-up phenomenon. In this work we study a modified form of the PKS equation that is natural for applications and for which global existence and uniqueness of solutions are easily seen to hold. Our focus here is instead on the study of the long time behavior through certain interacting particle systems. Under the so-called “quasi-stationary hypothesis” on the chemical field, the limit PDE reduces to a parabolic-elliptic system that is closely related to granular media equations whose time asymptotic properties have been extensively studied probabilistically through certain Lyapunov functions [17, 4, 9]. The modified PKS equation studied in the current work is a parabolic-parabolic system for which analogous Lyapunov function constructions are not available. A key challenge in the analysis is that the associated interacting particle system is not a Markov process as the interaction term depends on the whole history of the empirical measure. We establish, under suitable conditions, uniform in time convergence of the empirical measure of particle states to the solution of the PDE. We also provide uniform in time exponential concentration bounds for rate of the above convergence under additional integrability conditions. Finally, we introduce an Euler discretization scheme for the simulation of the interacting particle system and give error bounds that show that the scheme converges uniformly in time and in the size of the particle system as the discretization parameter approaches zero.

Mathematical analysis of cardiac electromechanics with physiological ionic model

This paper is concerned with the mathematical analysis of a coupled elliptic-parabolic system modeling the interaction between the propagation of electric potential coupled with general physiological ionic models and subsequent deformation of the cardiac tissue. A prototype system belonging to this class is provided by the electromechanical bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. The coupling between muscle contraction, biochemical reactions and electric activity is introduced with a so-called active strain decomposition framework, where the material gradient of deformation is split into an active (electrophysiology-dependent) part and an elastic (passive) one. We prove existence of weak solutions to the underlying coupled electromechanical bidomain model under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffu-sivities. The proof of the existence result, which constitutes the main thrust of this paper, is proved by means of a non-degenerate approximation system, the Faedo-Galerkin method, and the compactness method.

Gevrey regularity of mild solutions to the parabolic–elliptic system of drift-diffusion type in critical Besov spaces

This paper is devoted to studying analyticity of mild solutions to the parabolic–elliptic system of drift-diffusion type with small initial data in critical Besov spaces. More precisely, by using multilinear singular integrals and Fourier localization argument, we show that global-in-time mild solutions are Gevrey regular, i.e., (etΛ1v,etΛ1w)∈L˜t∞B˙p,r−2+np(Rn)∩L˜t1B˙p,rnp(Rn) for all 1<p<2n and 1≤r≤∞, where Λ1 is the Fourier multiplier whose symbol is given by |ξ|1=∑i=1n|ξi|. As a corollary, we obtain decay estimates of mild solutions in critical Besov spaces without using cumbersome recursive estimation of higher-order derivatives.

Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis

This paper concerns pattern formation in a class of reaction-advection-diffusion systems modeling the population dynamics of two predators and one prey. We consider the biological situation that both predators forage along the population density gradient of the preys which can defend themselves as a group. We prove the global existence and uniform boundedness of positive classical solutions for the fully parabolic system over a bounded domain with space dimension $N=1,2$ and for the parabolic- -parabolic-elliptic system over higher space dimensions. Linearized stability analysis shows that prey-taxis stabilizes the positive constant equilibrium if there is no group defense while it destabilizes the equilibrium otherwise. Then we obtain stationary and time-periodic nontrivial solutions of the system that bifurcate from the positive constant equilibrium. Moreover, the stability of these solutions is also analyzed in detail which provides a wave mode selection mechanism of nontrivial patterns for this strongly coupled system. Finally, we perform numerical simulations to illustrate and support our theoretical results.

On a Parabolic–Elliptic system with chemotaxis and logistic type growth

We consider a nonlinear PDEs system of two equations of Parabolic–Elliptic type with chemotactic terms. The system models the movement of a biological population “u” towards a higher concentration of a chemical agent “w” in a bounded and regular domain Ω⊂RN for arbitrary N∈N. After normalization, the system is as followsut−Δu=−div(umχ∇w)+μu(1−uα), in ΩT=Ω×(0,T),−Δw+w=uγ, in ΩT, for some positive constants m, χ, μ, α and γ, with positive initial datum u0 and Neumann boundary conditions.We study the range of parameters and constrains for which the solution exists globally in time. If eitherα>m+γ−1, orα=m+γ−1 and μ>Nα−22(m−1)+Nαχ, the solution exists globally in time. Moreover, ifα≥m+γ−1 and μ>2χ, and there exist positive constants u‾0 and u_0 such that 0<u_0≤u0≤u‾0<∞ we have that‖u−1‖L∞(Ω)+‖w−1‖L∞(Ω)→0 as t→∞.

Finite volume methods for a Keller–Segel system: discrete energy, error estimates and numerical blow-up analysis

We consider the finite volume approximation for a non-linear parabolic-elliptic system, which describes the aggregation of slime molds resulting from their chemotactic features, called a simplified Keller---Segel system. First, we present a linear finite volume scheme that satisfies both positivity and mass conservations, which are important features of the original system. We derive some inequalities on the discrete free energy. Then, under some assumptions on the regularity of solution, admissible mesh and a priori estimates of the discrete solution, we establish error estimates in $$L^p$$Lp norm with a suitable $$p>2$$p>2 for the two dimensional case. In the last part of this paper, we restrict our attention to the radially symmetric solution of chemotaxis system, and we derive some inequalities concerned with the blow-up phenomenon of numerical solution. Several numerical experiments are presented to verify the theoretical results.

Notes on a PDE system for biological network formation

We present new analytical and numerical results for the elliptic–parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transport networks. The model describes the pressure field using a Darcy’s type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. The analytical part extends the results of Haskovec et al. (2015) regarding the existence of weak and mild solutions to the whole range of meaningful relaxation exponents. Moreover, we prove finite time extinction or break-down of solutions in the spatially one-dimensional setting for certain ranges of the relaxation exponent. We also construct stationary solutions for the case of vanishing diffusion and critical value of the relaxation exponent, using a variational formulation and a penalty method.The analytical part is complemented by extensive numerical simulations. We propose a discretization based on mixed finite elements and study the qualitative properties of network structures for various parameter values. Furthermore, we indicate numerically that some analytical results proved for the spatially one-dimensional setting are likely to be valid also in several space dimensions.

Semi-implicit finite volume schemes for a chemotaxis-growth model

This paper is concerned with finite volume approximations for a nonlinear parabolic–elliptic system for chemotaxis-growth in Rd, d=2,3. This model describes a process of pattern formation by some chemotactic biological individuals. We present two schemes which make use of a semi-implicit time discretization and an upwind finite volume approximation. For both schemes, we prove existence, uniqueness and nonnegativity of the approximate solutions under some conditions on the time step, and we show (for one of the schemes) that the numerical solution converges to a corresponding weak solution for the studied model. Numerical simulations are performed in two dimensional spaces to demonstrate the efficiency of the schemes to capture the pattern formations and to verify our theoretical results.

Nonlinear semigroups generated by j-elliptic functionals

We generalise the theory of energy functionals used in the study of gradient systems to the case where the domain of definition of the functional cannot be embedded into the Hilbert space H on which the associated operator acts, such as when H is a trace space. We show that under weak conditions on the functional φ and the map j from the effective domain of φ to H, which in opposition to the classical theory does not have to be injective or even continuous, the operator on H naturally associated with the pair (φ,j) nevertheless generates a nonlinear semigroup of contractions on H. We show that this operator, which we call the j-subgradient of φ, is the (classical) subgradient of another functional on H, and give an extensive characterisation of this functional in terms of φ and j. In the case where H is an L2-space, we also characterise the positivity, L∞-contractivity and existence of order-preserving extrapolations to Lq of the semigroup in terms of φ and j. This theory is illustrated through numerous examples, including the p-Dirichlet-to-Neumann operator, general Robin-type parabolic boundary value problems for the p-Laplacian on very rough domains, and certain coupled parabolic–elliptic systems.

Degenerate two-phase porous media flow model with dynamic capillarity

In this paper, we investigate a degenerate elliptic–parabolic system which describes the flow of two incompressible, immiscible fluids in porous media and including dynamic effects in the phase-pressure difference. First, for the regularized case, the existence and uniqueness of the weak solution are obtained. Then we let the regularization parameter go to zero to show the existence of weak solutions under degenerate case.