Diffusion and reaction with monotone kinetics

Bandle, Catherine; Sperb, René P.; Stakgold, Ivar

doi:10.1016/0362-546x(84)90034-8

Cited by 72 publications

(46 citation statements)

References 9 publications

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“…One of the main results is that if n is a two-dimensional convex domain then N" is a convex set provided f satisfies (0. 3) f'(t)+{~\>O forO<t~l, i.e. every tangent line to y = f(t) intersects the line {t = l} in {y > O} (this condition is satisfied for some reaction-diffusion models).…”

Section: F(t) ~ T P As T ~O F( T) > 0 If T > 0mentioning

confidence: 99%

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The free boundary of a semilinear elliptic equation

Friedman¹,

Phillips²

1984

Trans. Amer. Math. Soc.

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ABSTRACT. The Dirichlet problem t1u = Af(u) in a domain n, u = I on an isis not monotone in general. The set {u = O} and the "free boundary" il{u = O} are studied. Sharp asymptotic estimates are established as A ~ 00. For suitable f, under the assumption that n is a two-dimensional convex domain, it is shown that {u = O} is a convex set. Analogous results are established also in the case where au/ap + JL( u -I) = 0 on an.

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Section: F(t) ~ T P As T ~O F( T) > 0 If T > 0mentioning

confidence: 99%

“…which arise in reaction-diffusion [2,3,7,17]. The condition (2.1) is satisfied for some range of the parameters {3, y. and u > 0 in n.…”

Section: ) Notice That C(x)mentioning

confidence: 99%

The free boundary of a semilinear elliptic equation

Friedman¹,

Phillips²

1984

Trans. Amer. Math. Soc.

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“…For a physical application, we consider the single, irreversible, steady-state reaction studied in [1]. There the scalar problem for the concentration is formulated as is given in terms of the value of u at x and m = minu(x) when the average curvature of the boundary is nonnegative.…”

Section: Boundsmentioning

confidence: 99%

Some Maximum Principles in Semilinear Elliptic Equations

Schaefer¹

1986

Proceedings of the American Mathematical Society

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ABSTRACT.We develop maximum principles for functions defined on the solutions to a class of semilinear, second order, uniformly elliptic partial differential equations. These principles are related to recent theorems of Protter and Protter and Weinberger and to a technique initiated by Payne for the determination of gradient bounds on the solution of the equation.1. Introduction. In [3] Payne introduced a technique, which utilizes a maximum principle for a function defined on solutions to an elliptic partial differential equation, in order to obtain bounds for the gradient of the solution of the relèvent differential equation. Several authors have contributed to the growing literature developing this technique. In their work (see the references cited here, especially [7], and the references therein), the authors seek estimates on the solution, the gradient of the solution, or other quantities of physical importance and/or extend the method to more general elliptic or parabolic differential equations. Early in the development of this method, the results were obtained when the principal part of the elliptic equation was the Laplace operator.Sperb [6] was the first to extend the results of Payne to a second order uniformly elliptic equation of the form

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“…Given 2e~ + and pe(0, 1), consider the problem: find u such that The semilinear problem (1.1) arises in the study of an isothermal reaction, u being the concentration of the reactant (see Bandle et al 1984). By considering the associated minimization problem: minimize for v e H~ (f2) the functional (1.3a)…”

Section: Introductionmentioning

confidence: 99%

Finite element approximation of a model reaction ? diffusion problem with a non-Lipschitz nonlinearity

Barrett

Shanahan

1991

Numer. Math.

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Summary. Given 2>0 and p~(0, 1), we consider the following problem: findwhere ~r~c~-~2 is a smooth convex domain. We prove optimal H 1 and L ~ error bounds for the standard continuous piecewise linear Galerkin finite element approximation. In addition we analyse a more practical approximation using numerical integration on the nonlinear term. Finally we consider a modified nonlinear SOR algorithm, which is shown to be globally convergent, for solving the algebraic system derived from the more practical approximation.

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Diffusion and reaction with monotone kinetics

Cited by 72 publications

References 9 publications

The free boundary of a semilinear elliptic equation

The free boundary of a semilinear elliptic equation

Some Maximum Principles in Semilinear Elliptic Equations

Finite element approximation of a model reaction ? diffusion problem with a non-Lipschitz nonlinearity

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