An application of the monotone iterative method to a combustion problem in porous media

Mota, J. C. da; Santos, Marcelo M.

doi:10.1016/j.nonrwa.2010.09.012

Cited by 4 publications

(4 citation statements)

References 5 publications

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“…the system in (3.9) without delta). (See Lemma 2 in [3].) Indeed, it is obvious that û = (0, 0) is a lower solution (in fact, a solution) to this system, since f i (0, 0) = 0.…”

Section: Proof Of Theoremmentioning

confidence: 98%

“…And as a corollary we obtain 2 If g is a bounded function defined in R, g ∞ := sup x∈R |g(x)|. 3 If T < ∞ and the function u i is defined and continuous in R × [0, T ], obviously we can extend the inequality 0 ≤ u i (x, t) ≤ ϕ(t) to t = T . 4 Here the term "loc" stands for "locally" in time, i.e.…”

Section: Introductionmentioning

confidence: 97%

“…We denote by ϕ the "upper solution" ϕ(t) = (M + β)e αt − β for the Cauchy problem (1.1), (1.6) with given y i , satisfying 0 ≤ y i ≤ y i,0 ∞ , where 2 M = max i=1,2 u i,0 ∞ , α = max i=1,2 { Aibi yi,0 ∞ ai } and β = max i=1,2 { di Aibi }, and, for 0 < T ≤ ∞, we denote by 0, ϕ T the sector (set) of vector functions u = (u 1 , u 2 ) : R × [0, T ) → R 2 such that 0 ≤ u i (x, t) ≤ ϕ(t) (for i = 1, 2 and) for all (x, t) ∈ R × [0, T ). 3 It is easy to check that the pair of (vector) functions û := (0, 0) and ũ := (ϕ, ϕ) is an ordered pair (ordered in the sense that ûi ≤ ũi ) of lower and upper solutions to the system (1.1) [3, Lemma 2]. See Section 3.1, p. 22, for details.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Cauchy problem for a combustion model in a porous medium with two layers

Mota

Santos

2017

Monatsh Math

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We prove the existence of a global solution to the Cauchy problem for a nonlinear reaction-diffusion system coupled with a system of ordinary differential equations. The system models the propagation of a combustion front in a porous medium with two layers, as derived by J. C. da Mota and S. Schecter in Combustion fronts in a porous medium with two layers, Journal of Dynamics and Differential Equations, 18(3) (2006). For the particular case, when the fuel concentrations in both layers are known functions, the Cauchy problem was solved by J. C. da Mota and M. M. Santos in An application of the monotone iterative method to a combustion problem in porous media, Nonlinear Analysis: Real World Application, 12 (2010). For the full system, in which the fuel concentrations are also unknown functions, we construct an iterative scheme that contains a sequence which converges to a solution of the system, locally in time, under the conditions that the initial data are Hölder continuous, bounded and nonnegative functions. We also show the existence of a global solution, if the initial date are additionally in the Lebesgue space L p , for some p ∈ (1, ∞). Our proof of the local existence relies on a careful analysis on the construction of the fundamental solution for parabolic equations obtained by the parametrix method. In particular, we show the continuous dependence of the fundamental solution for parabolic equations with respect to the coefficients of the equations. To obtain the global existence, we employ the "method of auxiliary functions" as used by O. A. Oleinik and S. N. Kruzhkov in Quasi-linear second-order parabolic equations with many independent variables, Russian Mathematical Surveys, 16(5) (1961). Furthermore, for a broad class of reaction-diffusion systems we show that the non negative quadrant is a positively invariant region, and, as a consequence, that classical solutions of similar systems, with the reactions functions being non decreasing in one unknown and semi-lipscthitz continuous in the other, are bounded by lower and upper solutions for any positive time if so they are at time zero.

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“…the system in (3.9) without delta). (See Lemma 2 in [3].) Indeed, it is obvious that û = (0, 0) is a lower solution (in fact, a solution) to this system, since f i (0, 0) = 0.…”

Section: Proof Of Theoremmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Cauchy problem for a combustion model in a porous medium with two layers

Mota

Santos

2017

Monatsh Math

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“…Mais recentemente, alguns autores naárea de Matemática Aplicada vem trabalhando com modelos de combustão em meios porosos simplificados, de tal modo que são passíveis de serem analisados com as teorias matemáticas hoje disponíveis. Como exemplos, citamos [1,2,[5][6][7][8][9][10][11][12]. No presente trabalho, estudamos a existência e unicidade de solução global no tempo para um problema de valor inicial e de contorno associado a um sistema de reação-difusão-convecção, com n equações parabólicas não lineares, acoplado com n 2 equações diferenciais ordinárias.…”

Section: Introductionunclassified

Existência e Unicidade de Solução para um Problema de Combustão num Meio Poroso com n Camadas

Mota¹,

Batista²

2020

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Neste trabalho estudamos as soluções clássicas para um sistema parabólico de equações de reação-difusão-convecção, acoplado a um sistema de equações diferenciais ordinárias, com condições iniciais e de contorno num domínio limitado. O sistema acoplado modela a propagação de uma frente de combustão através de um meio poroso com n camadas paralelas, onde as variáveis dependentes são as temperaturas e as concentrações de combustível em cada camada. O modeloé uma generalização do modelo de duas camadas estudado em [6]. Usando o Método Iterativo Monótono, provamos a existência e unicidade de solução global no tempo para o caso particular, em que as concentrações de combustível em cada camada são funções conhecidas. Palavras-chave. Combustão, Sistema de reação-difusão-convecção, Método iterativo monótono.

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Monotone iterative method of upper and lower solutions applied to a multilayer combustion model in porous media

Batista¹,

Mota

2021

Nonlinear Analysis: Real World Applications

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An application of the monotone iterative method to a combustion problem in porous media

Cited by 4 publications

References 5 publications

The Cauchy problem for a combustion model in a porous medium with two layers

The Cauchy problem for a combustion model in a porous medium with two layers

Existência e Unicidade de Solução para um Problema de Combustão num Meio Poroso com n Camadas

Monotone iterative method of upper and lower solutions applied to a multilayer combustion model in porous media

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