Global Existence of Solutions to Reaction-Diffusion Systems with Mass Transport Type Boundary Conditions
14 Dec 2016-Siam Journal on Mathematical Analysis (Society for Industrial and Applied Mathematics)-Vol. 48, Iss: 6, pp 4202-4240
TL;DR: Local well-posedness and global existence of solutions for reaction-diffusion systems are established using classical potential theory and linear estimates for initial boundary value problems.
Abstract: We consider a reaction-diffusion system where some components react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those on the boundary through mass transport. We establish local well-posedness and global existence of solutions for these systems using classical potential theory and linear estimates for initial boundary value problems.
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01 Jan 2016
TL;DR: In this article, the authors present a broad overview of the application of the Hopf maximum principle in the study of partial differential equations (PDE) with a focus on elliptic and hyperbolic solutions.
Abstract: This book is devoted to the study of maximum principles in partial differential equations. I t contains a wealth of material much of which is presented for the first time in a book form. An attractive feature of the book is that it is completely elementary and thus accessible to a wide audience of readers. The book has four chapters. Chapter I deals with the one dimensional maximum principle. The discussion of this very simple model of a maximum principle forms a good introduction to the general theory. Various applications of the principle are given to show that it is a very useful tool even in the study of ordinary differential equations. As an example, it is shown that many oscillation and comparison results in the Sturm-Liouville theory could be deduced most easily by a maximum principle argument. The proper discussion of maximum principles in partial differential equations begins in Chapter II . This chapter, which is the backbone of the book, is devoted to elliptic equations. The material covered in this chapter includes the E. Hopf maximum principle and its generalizations; the Phragmèn-Lindelöf principle for solutions of elliptic equations; Serrin's version of the Harnack inequality for solutions of general elliptic equations in two variables (this is probably the most difficult result discussed in the book) ; various versions of the Hadamard three circles theorems for solutions of elliptic equations; applications of the maximum principle to nonlinear equations and to problems of fluid flow. Chapter III is devoted to parabolic equations. The plan of this chapter parallels that of Chapter II . The topics discussed include the L. Nirenberg strong maximum principle; a three curves theorem with an interesting application to the Tikhonov uniqueness theorem; a Phragmèn-Lindelöf principle for parabolic equations with applications to uniqueness results; nonlinear operators; a maximum principle for certain parabolic systems. The fourth and the last chapter is devoted to hyperbolic equations. The results in this chapter are somewhat special since a maximum principle in the proper sense does not hold for solutions of hyperbolic equations. Nevertheless, solutions of certain hyperbolic equations
207 citations
TL;DR: This study presents a three-dimensional generalisation of this mathematical framework through the maturing theory of coupled bulk-surface semilinear partial differential equations in which protein compartmentalisation becomes natural.
68 citations
Cites methods from "Global Existence of Solutions to Re..."
...See [54] and in particular Corollary 3....
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...It should be noted that the well-posedness and the global existence of solutions for the general bulk-surface reaction-diffusion system of k bulk and m surface variables was studied by Sharma and Morgan in [54]....
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Posted Content•
TL;DR: A coupled bulk-surface system of partial differential equations with nonlinear coupling modelling receptor-ligand dynamics that proves the existence and uniqueness of solutions and convergence to limiting problems which take the form of free boundary problems posed on the cell surface is considered.
Abstract: We consider a coupled bulk-surface system of partial differential equations with nonlinear coupling modelling receptor-ligand dynamics. The model arises as a simplification of a mathematical model for the reaction between cell surface resident receptors and ligands present in the extra-cellular medium. We prove the existence and uniqueness of solutions. We also consider a number of biologically relevant asymptotic limits of the model. We prove convergence to limiting problems which take the form of free boundary problems posed on the cell surface. We also report on numerical simulations illustrating convergence to one of the limiting problems as well as the spatio-temporal distributions of the receptors and ligands in a realistic geometry.
22 citations
TL;DR: In this paper, a certain class of coupled bulk-surface reaction-drift-diffusion systems arising in the modeling of signalling networks in biological cells is analyzed and the existence of weak and classical solutions for reaction terms with at most linear growth is shown.
Abstract: We analyze a certain class of coupled bulk–surface reaction–drift–diffusion systems arising in the modeling of signalling networks in biological cells. The coupling is by a nonlinear Robin-type boundary condition for the bulk variable and a corresponding source term on the cell boundary. For reaction terms with at most linear growth and under different regularity assumptions on the data we prove the existence of weak and classical solutions. In particular, we show that solutions grow at most exponentially with time. Furthermore, we rigorously derive an asymptotic reduction to a non-local reaction–drift–diffusion system on the membrane in the fast-diffusion limit.
17 citations
TL;DR: In this paper, two full models were derived from reaction-diffusion equations in the bulk and the thin membrane, respectively, with two types of reasonable transmission conditions linking the two, and in the limit of δ → 0, they obtained two effective models, with one being a bulk-surface model and the other being the dynamical boundary value problem model, from which the surface density of the other substance was recovered.
12 citations
References
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Book•
11 Feb 1992
TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
Abstract: 1 Generation and Representation.- 1.1 Uniformly Continuous Semigroups of Bounded Linear Operators.- 1.2 Strongly Continuous Semigroups of Bounded Linear Operators.- 1.3 The Hille-Yosida Theorem.- 1.4 The Lumer Phillips Theorem.- 1.5 The Characterization of the Infinitesimal Generators of C0 Semigroups.- 1.6 Groups of Bounded Operators.- 1.7 The Inversion of the Laplace Transform.- 1.8 Two Exponential Formulas.- 1.9 Pseudo Resolvents.- 1.10 The Dual Semigroup.- 2 Spectral Properties and Regularity.- 2.1 Weak Equals Strong.- 2.2 Spectral Mapping Theorems.- 2.3 Semigroups of Compact Operators.- 2.4 Differentiability.- 2.5 Analytic Semigroups.- 2.6 Fractional Powers of Closed Operators.- 3 Perturbations and Approximations.- 3.1 Perturbations by Bounded Linear Operators.- 3.2 Perturbations of Infinitesimal Generators of Analytic Semigroups.- 3.3 Perturbations of Infinitesimal Generators of Contraction Semigroups.- 3.4 The Trotter Approximation Theorem.- 3.5 A General Representation Theorem.- 3.6 Approximation by Discrete Semigroups.- 4 The Abstract Cauchy Problem.- 4.1 The Homogeneous Initial Value Problem.- 4.2 The Inhomogeneous Initial Value Problem.- 4.3 Regularity of Mild Solutions for Analytic Semigroups.- 4.4 Asymptotic Behavior of Solutions.- 4.5 Invariant and Admissible Subspaces.- 5 Evolution Equations.- 5.1 Evolution Systems.- 5.2 Stable Families of Generators.- 5.3 An Evolution System in the Hyperbolic Case.- 5.4 Regular Solutions in the Hyperbolic Case.- 5.5 The Inhomogeneous Equation in the Hyperbolic Case.- 5.6 An Evolution System for the Parabolic Initial Value Problem.- 5.7 The Inhomogeneous Equation in the Parabolic Case.- 5.8 Asymptotic Behavior of Solutions in the Parabolic Case.- 6 Some Nonlinear Evolution Equations.- 6.1 Lipschitz Perturbations of Linear Evolution Equations.- 6.2 Semilinear Equations with Compact Semigroups.- 6.3 Semilinear Equations with Analytic Semigroups.- 6.4 A Quasilinear Equation of Evolution.- 7 Applications to Partial Differential Equations-Linear Equations.- 7.1 Introduction.- 7.2 Parabolic Equations-L2 Theory.- 7.3 Parabolic Equations-Lp Theory.- 7.4 The Wave Equation.- 7.5 A Schrodinger Equation.- 7.6 A Parabolic Evolution Equation.- 8 Applications to Partial Differential Equations-Nonlinear Equations.- 8.1 A Nonlinear Schroinger Equation.- 8.2 A Nonlinear Heat Equation in R1.- 8.3 A Semilinear Evolution Equation in R3.- 8.4 A General Class of Semilinear Initial Value Problems.- 8.5 The Korteweg-de Vries Equation.- Bibliographical Notes and Remarks.
11,637 citations
Book•
31 Dec 1969
TL;DR: In this article, the authors considered a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtained parameter dependent time decay estimates of the difference between the solutions of the solution of a quasi-linear parabolic equation and the corresponding linear parabolic equations.
Abstract: linear and quasi linear equations of parabolic type by o a ladyzhenskaia 1968 american mathematical society edition in english, note citations are based on reference standards however formatting rules can vary widely between applications and fields of interest or study the specific requirements or preferences of your reviewing publisher classroom teacher institution or organization should be applied, we consider a hyperbolicparabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtain parameter dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation, pris 1899 kr hftad 1968 tillflligt slut bevaka linear and quasi linear equations of parabolic type s fr du ett mejl nr boken gr att kpa igen, then u x t solves the following system of quasilinear parabolic pde where y is the infinitesimal operator generated by the diffusion process y a particular case is that of linear one dimensional backward equation where f does not contain q in this case the corresponding system of equation becomes a linear parabolic pde, we consider linear parabolic equations of second order in a sobolev space setting we obtain existence and uniqueness results for such equations on a closed two dimensional manifold with minimal assumptions about the regularity of the coefficients of the elliptic operator, linear equations of the second order of parabolic type a m il in a s kalashnikov and o a oleinik the solvability of mixed problems for hyperbolic and parabolic equations v a il in quasi linear elliptic equations and variational problems with many independent variableso a ladyzhenskaya and n n ural tseva, the first boundary value problem for a linear second order parabolic equation is studied under the assumption that the inhomogeneous term is continuous in space and time and hlder continuous only with respect to the space variables linear and quasilinear equations of parabolic type transl math monographs 23 providence amer math, collapse in finite time is established for part of the solutions of certain classes of quasilinear equations of parabolic and hyperbolic types the linear part of which has general form certain hyperbolic equations having l m pairs belong to these classes, find helpful customer reviews and review ratings for linear and quasi linear equations of parabolic type at amazon com read honest and unbiased product
7,118 citations
"Global Existence of Solutions to Re..." refers background in this paper
...1 in Chapter 4 of [19] to get the existence of a Hölder continuous solution to system (3....
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...1 in [19] to obtain an estimate for φ in W 2,1 p (Ω̃× (0, T ))....
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...From Sobolev embedding (see [11], [19]), u0, v0 are bounded functions....
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...342, section 9 of Chapter 4, in [19], and the comparison principle....
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...Since p > n + 1, W 2,1 p (Ω̃ × (0, T )) embeds continuously into the space of Hölder continuous functions (see [19])....
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Book•
01 Feb 1981
TL;DR: The neighborhood of an invariant manifold near an equilibrium point is a neighborhood of nonlinear parabolic equations in physical, biological and engineering problems as mentioned in this paper, where the neighborhood of a periodic solution is defined by the invariance of the manifold.
Abstract: Preliminaries.- Examples of nonlinear parabolic equations in physical, biological and engineering problems.- Existence, uniqueness and continuous dependence.- Dynamical systems and liapunov stability.- Neighborhood of an equilibrium point.- Invariant manifolds near an equilibrium point.- Linear nonautonomous equations.- Neighborhood of a periodic solution.- The neighborhood of an invariant manifold.- Two examples.
6,056 citations
Book•
01 Jan 1964
4,652 citations
"Global Existence of Solutions to Re..." refers background in this paper
...From Sobolev embedding (see [11], [19]), u0, v0 are bounded functions....
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