A blow-up criterion for a degenerate parabolic problem due to a concentrated nonlinear source

Chan, C. Y.; Boonklurb, Ratinan

doi:10.1090/s0033-569x-07-01082-9

Cited by 6 publications

(5 citation statements)

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“…It can be shown that, the positive solution of (3.3)-(3.4) exists and is unique [7,8]. Furthermore, for any fixed ratio r = a/b, there exists a critical value a * such that if a a * , then there is a quenching time T a , T a < ∞, and…”

Section: Decompositions For Singular Reaction-diffusion Problemsmentioning

confidence: 99%

“…The phenomenon is called quenching, which is different from traditional blowup problems [1][2][3][4][5][6][7][8][9]21,38,54]. A particular value of a * depends on the structure of the model problem considered.…”

Section: Decompositions For Singular Reaction-diffusion Problemsmentioning

confidence: 99%

“…Discussion of the existence and uniqueness of its solutions can be found in Refs. [6][7][8]. Classical numerical algorithms with large uniform spatial and temporal grids may fail to reproduce quenching phenomena in the numerical solution since length scale of the singularity may be less than a step size.…”

Section: Decompositions For Singular Reaction-diffusion Problemsmentioning

confidence: 99%

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Adaptive decomposition finite difference methods for solving singular problems—A review

Sheng

2009

Front. Math. China

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Decomposition, or splitting, finite difference methods have been playing an important role in the numerical solution of nonsingular differential equation problems due to their remarkable efficiency, simplicity, and flexibility in computations as compared with their peers. Although the numerical strategy is still in its infancy for solving singular differential equation problems arising from many applications, explorations of the next generation decomposition schemes associated with various kinds of adaptations can be found in many recent publications. The novel approaches have been proven to be highly effective and reliable in operations. In this article, we will focus on some of the latest developments in the area. Key comments and discussion will be devoted to two particularly interesting issues in the research, that is, direct solutions of degenerate singular reactiondiffusion equations and nonlinear sine-Gordon wave equations. Numerical experiments with simulated demonstrations will be given.

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Section: Decompositions For Singular Reaction-diffusion Problemsmentioning

confidence: 99%

Section: Decompositions For Singular Reaction-diffusion Problemsmentioning

confidence: 99%

Section: Decompositions For Singular Reaction-diffusion Problemsmentioning

confidence: 99%

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Adaptive decomposition finite difference methods for solving singular problems—A review

Sheng

2009

Front. Math. China

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“…When α = 1, the operator L α becomes the classical heat operator L 1 u = u t − u x x , and the effects of the nonlinear concentrated source on the semilinear problems were studied by many mathematicians. For example, Chan, Chan and Boonklurb, Chan and Tian, and Chan and Tragoonsirisak studied the behavior of the continuous solution of the problems for different types of parabolic operators including semilinear, degenerate parabolic operators with different nonlinear forcing term in different domains.…”

Section: Introductionmentioning

confidence: 99%

Blow‐up behavior of the Solution for the problem in a subdiffusive mediums

Liu

2018

Math Methods in App Sciences

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“…By analyzing its corresponding nonlinear Volterra equation at the site of the concentrated source, they showed that blowup can be prevented by locating the nonlinear source sufficiently close to the boundary of the domain. Chan and Boonklurb [1] studied the following degenerate semilinear parabolic first initial-boundary value problem with a concentrated source at b ∈ (0, 1) on a bounded domain:…”

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confidence: 99%

Blow-up criteria for a parabolic problem due to a concentrated nonlinear source on a semi-infinite interval

Chan

Treeyaprasert

2011

Quart. Appl. Math.

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Let α, b and T be positive numbers, D = (0, ∞),D = [0, ∞), and Ω = D × (0, T ]. This article studies the first initial-boundary value problem, u t − u xx = αδ(x − b)f (u(x, t)) in Ω, u(x, 0) = ψ(x) onD, u(0, t) = 0 = lim x→∞ u(x, t) for 0 < t ≤ T, where δ (x) is the Dirac delta function, and f and ψ are given functions. We assume that f (0) ≥ 0, f (u) and its derivatives f (u) and f (u) are positive for u > 0, and ψ(x) is nontrivial, nonnegative and continuous such that ψ (0) = 0 = lim x→∞ ψ (x), and ψ + αδ(x − b)f (ψ) ≥ 0 in D. It is shown that if u blows up, then it blows up in a finite time at the single point b only. A criterion for u to blow up in a finite time and a criterion for u to exist globally are given. It is also shown that there exists a critical position b * for the nonlinear source to be placed such that no blowup occurs for b ≤ b * , and u blows up in a finite time for b > b *. This also implies that u does not blow up in infinite time. The formula for computing b * is also derived. For illustrations, two examples are given.

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A blow-up criterion for a degenerate parabolic problem due to a concentrated nonlinear source

Abstract: Abstract. Let q, a, b, and T be real numbers with q ≥ 0, a > 0, 0 < b < 1, and T > 0. This article studies the following degenerate semilinear parabolic first initial-boundary value problem,

Cited by 6 publications

References 5 publications

Adaptive decomposition finite difference methods for solving singular problems—A review

Adaptive decomposition finite difference methods for solving singular problems—A review

Blow‐up behavior of the Solution for the problem in a subdiffusive mediums

Blow-up criteria for a parabolic problem due to a concentrated nonlinear source on a semi-infinite interval

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