Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions
01 Jan 2021-Communications on Pure and Applied Analysis (American Institute of Mathematical Sciences)-Vol. 20, Iss: 3, pp 955
TL;DR: In this paper, the authors consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions and establish the existence of component wise non-negative global solutions which are uniformly bounded in the sup norm.
Abstract: We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions. Under reasonable hypotheses, we establish the existence of component wise non-negative global solutions which are uniformly bounded in the sup norm.
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TL;DR: In this paper, the authors studied the global existence of classical solutions to volume-surface reaction-diffusion systems with control of mass and showed that the solution is bounded uniformly in time by studying the system on each time-space cylinder of unit size.
Abstract: We study the global existence of classical solutions to volume-surface reaction-diffusion systems with control of mass. Such systems appear naturally from modeling evolution of concentrations or densities appearing both in a volume domain and its surface, and therefore have attracted considerable attention. Due to the characteristic volume-surface coupling, global existence of solutions to general systems is a challenging issue. In particular, the duality method, which is powerful in dealing with mass conserved systems in domains, is not applicable on its own. In this paper, we introduce a new family of $L^p$-energy functions and combine them with a suitable duality method for volume-surface systems, to ultimately obtain global existence of classical solutions under a general assumption called the \textit{intermediate sum condition}. For systems that conserve mass, but do not satisfy this condition, global solutions are shown under a quasi-uniform condition, that is, under the assumption that the diffusion coefficients are close to each other. In the case of mass dissipation, we also show that the solution is bounded uniformly in time by studying the system on each time-space cylinder of unit size, and showing that the solution is sup-norm bounded independently of the cylinder. Applications of our results include global existence and boundedness for systems arising from membrane protein clustering or activation of Cdc42 in cell polarization.
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TL;DR: In this paper , the authors studied the global existence of classical solutions to volume-surface reaction-diffusion systems with control of mass and showed that the solution is bounded uniformly in time by studying the system on each time-space cylinder of unit size.
Abstract: We study the global existence of classical solutions to volume–surface reaction–diffusion systems with control of mass. Such systems appear naturally from modeling evolution of concentrations or densities appearing both in a volume domain and its surface, and therefore have attracted considerable attention. Due to the characteristic volume–surface coupling, global existence of solutions to general systems is a challenging issue. In particular, the duality method, which is powerful in dealing with mass conserved systems in domains, is not applicable on its own. In this paper, we introduce a new family of [Formula: see text]-energy functions and combine them with a suitable duality method for volume–surface systems, to ultimately obtain global existence of classical solutions under a general assumption called the intermediate sum condition. For systems that conserve mass, but do not satisfy this condition, global solutions are shown under a quasi-uniform condition, that is, under the assumption that the diffusion coefficients are close to each other. In the case of mass dissipation, we also show that the solution is bounded uniformly in time by studying the system on each time-space cylinder of unit size, and showing that the solution is sup-norm bounded independently of the cylinder. Applications of our results include global existence and boundedness for systems arising from membrane protein clustering or activation of Cdc42 in cell polarization.
1 citations
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TL;DR: In this paper, the existence of component-wise non-negative global solutions for reaction-diffusion systems is established using Lyapunov functional and duality arguments, where components diffuse inside the domain and react on the surface through mass transport type boundary conditions on an evolving domain.
Abstract: We consider reaction-diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions on an evolving domain. Using Lyapunov functional and duality arguments, we establish the existence of component-wise non-negative global solutions.
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TL;DR: In this paper , the existence of component wise non-negative global solutions for reaction diffusion systems is established using a Lyapunov functional and duality argument. But the authors do not consider the nonnegative global solution in this paper.
Abstract: <p style='text-indent:20px;'>We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions on an evolving domain. Using a Lyapunov functional and duality arguments, we establish the existence of component wise non-negative global solutions.</p>
TL;DR: In this article, the existence of component wise non-negative global solutions for reaction diffusion systems is established using a Lyapunov functional and duality arguments, and the authors show that these solutions can be obtained in a continuous domain.
Abstract: We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions on an evolving domain. Using a Lyapunov functional and duality arguments, we establish the existence of component wise non-negative global solutions.
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01 Jan 1941
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Abstract: Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear partial differential equations: Sobolev spaces Second-order elliptic equations Linear evolution equations Part III: Theory for nonlinear partial differential equations: The calculus of variations Nonvariational techniques Hamilton-Jacobi equations Systems of conservation laws Appendices Bibliography Index.
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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
TL;DR: In this article, the authors describe the state of the art on the question of global existence of solutions to reaction-diffusion systems for which two main properties hold: on one hand, the positivity of the solutions is preserved for all time; on the other hand the total mass of the components is uniformly controlled in time.
Abstract: The goal of this paper is to describe the state of the art on the question of global existence of solutions to reaction-diffusion systems for which two main properties hold: on one hand, the positivity of the solutions is preserved for all time; on the other hand, the total mass of the components is uniformly controlled in time. This uniform control on the mass (or – in mathematical terms- on the L1-norm of the solution) suggests that no blow up should occur in finite time. It turns out that the situation is not so simple. This explains why so many partial results in different directions are found in the literature on this topic, and why also the general question of global existence is still open, while lots of systems arise in applications with these two natural properties. We recall here the main positive and negative results on global existence, together with many references, a description of the still open problems and a few new results as well.
272 citations