Existence of solution for the problem with a concentrated source in a subdiffusive medium

Liu, H. Terence; Huang, Weicheng

doi:10.1063/1.5020476

Cited by 2 publications

(5 citation statements)

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“…In this section, we investigate the nonexistence of the solution u of problems and . Recall from the study in the existence of the integral solution by Liu and Huang, if solution u exists in a finite time [0, t b ) only, then u is unbounded in ( − ∞ , ∞ ) × [0, t b ). We recall the following definition of blow‐up behavior:…”

Section: Resultsmentioning

confidence: 99%

“…In this section, we investigate the nonexistence of the solution u of problems (1) and (2). Recall from the study in the existence of the integral solution (4) by Liu and Huang, 8…”

Section: Resultsmentioning

confidence: 99%

“…Different from Olmstead and Roberts, recently Chan and Liu studied the initial boundary problem

\{\begin{matrix} L_{α} u false(x, t false) = δ false(x - b false) f false(u false(x, t false) false) in false(0, a false) \times false(0, T false], \\ u false(x, 0 false) = ϕ false(x false) on false[0, a false], \\ u false(0, t false) = 0 = u false(a, t false) for 0 < t \leq T . \end{matrix}

They showed the existence and nonexistence of the continuous solution u of the fractional differential equation in finite spacial domain [0, a ]. Liu and Huang extended these results to infinite domain. They showed properties of the integral representation form of the solution u ( x , t ) of problems and .…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, by using the Green's function G α ( x , t − τ ; ξ ), Liu and Huang studied the integral solution of problems and ,

u false(x, t false) = true \int_{- \infty}^{\infty} G_{α} false(x, t; ξ false) ϕ false(ξ false) d ξ + true \int_{0}^{t} G_{α} false(x, t - τ; b false) f false(u false(b, τ false) false) d τ,

and the existence of the solution of the integral equation is given. Furthermore, they showed that this integral solution is the solution of problems and and gave the following results.…”

Section: Introductionmentioning

confidence: 99%

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

Liu

2018

Math Methods in App Sciences

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Section: Resultsmentioning

confidence: 99%

“…In this section, we investigate the nonexistence of the solution u of problems (1) and (2). Recall from the study in the existence of the integral solution (4) by Liu and Huang, 8…”

Section: Resultsmentioning

confidence: 99%

“…Different from Olmstead and Roberts, recently Chan and Liu studied the initial boundary problem

\{\begin{matrix} L_{α} u false(x, t false) = δ false(x - b false) f false(u false(x, t false) false) in false(0, a false) \times false(0, T false], \\ u false(x, 0 false) = ϕ false(x false) on false[0, a false], \\ u false(0, t false) = 0 = u false(a, t false) for 0 < t \leq T . \end{matrix}

Section: Introductionmentioning

confidence: 99%

“…More precisely, by using the Green's function G α ( x , t − τ ; ξ ), Liu and Huang studied the integral solution of problems and ,

u false(x, t false) = true \int_{- \infty}^{\infty} G_{α} false(x, t; ξ false) ϕ false(ξ false) d ξ + true \int_{0}^{t} G_{α} false(x, t - τ; b false) f false(u false(b, τ false) false) d τ,

and the existence of the solution of the integral equation is given. Furthermore, they showed that this integral solution is the solution of problems and and gave the following results.…”

Section: Introductionmentioning

confidence: 99%

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

Liu

2018

Math Methods in App Sciences

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“…And hence those problems are denoted as subdiffusive problem, or problem in a sub diffusive medium. In term of the Green's function 𝐺 𝛼 (𝑥, 𝑡 − 𝜏; 𝜉) , by considering the integral representation form, it is capable to prove the existence of a continuous solution 𝑢 of the sub diffusive differential equation in domain [0, 𝑎], and later on Liu and Huang (2018) gave similar results in to infinite domain.…”

Section: Literature Reviewmentioning

confidence: 99%

Quenching Behavior of the Solution for the Problems With Sequential Concentrated Sources

Liu

2022

mijst

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This article studies the diffusion problems with a concentrated source which is provided at a sequential time steps in 1 dimensional space. The problems are considered for both Gaussian and fractional diffusion operators. For the fractional diffusion case, Riemann-Liouville operator with fractional order is used to describe the model with diffusion rate slower than normal time scale, which is known as sub diffusive problems. Due to this sub diffusive property, the existence and nonexistence behavior of the solution will be studied. Since the forcing term will experience a concentrated source at a sequence of time steps, the frequency, the time difference and strength of the source may affect the growth rate of the solution. Criteria for these effects which may cause for the quenching behavior of the solution will be given. The existence of the solution is investigated. The monotone behavior in spatial will be given. The quenching behavior of the solution will be studied. The location of the quenching point will be discussed.

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Existence of solution for the problem with a concentrated source in a subdiffusive medium

Cited by 2 publications

References 14 publications

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

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

Quenching Behavior of the Solution for the Problems With Sequential Concentrated Sources

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