Behavior of free boundaries of blow-up solutions to one-phase stefan problems

Aiki, Toyohiko

doi:10.1016/0362-546x(94)00311-5

Cited by 9 publications

(18 citation statements)

References 5 publications

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“…Throughout this paper, we suppose p > 1, s 0 > 0 and u 0 ∈ C 1 ([0, s 0 ]), u 0 0, with (u 0 ) x (0) = u 0 (s 0 ) = 0. It is well known that there exists a unique, maximal in time, classical solution (u, s) of (SP), which satisfies u 0 and s 0 (see [1,4]). † Email: fila@fmph.uniba.sk ‡ Also at: Laboratoire de Mathématiques Appliquées, UMR CNRS 7641, Université de Versailles, 45 avenue des Etats-Unis, 78035 Versailles, France.…”

Section: (Sp)mentioning

confidence: 99%

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Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem

Фила¹,

Souplet²

2001

Interfaces Free Bound.

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We consider a one-phase Stefan problem for the heat equation with a superlinear reaction term. It is known from a previous work (Ghidouche, Souplet, & Tarzia [5]) that all global solutions are bounded and decay uniformly to 0. Moreover, it was shown in Ghidouche, Souplet, & Tarzia [5] that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial, and it was also proved that small data solutions behave like (i). Here we prove that there exist global solutions with slow decay and unbounded free boundary, i.e. of type (ii). Also, we establish uniform a priori estimates for all global solutions. Moreover, we provide a correction to an error in the proof of decay from Ghidouche, Souplet, & Tarzia [5].

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Section: (Sp)mentioning

confidence: 99%

“…Earlier results on blowup for problem (SP) can be found in [1,7], where the shape of some blowup solutions was studied. On the other hand, the existence of global solutions to (SP) was studied in [2,3].…”

Section: (Sp)mentioning

confidence: 99%

Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem

Фила¹,

Souplet²

2001

Interfaces Free Bound.

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“…There are results concerned with behavior of blow-up solutions to one-phase Stefan problems (cf. [2,3]). Besides, we know the following result concerned with global existence for solutions of Cauchy problems for equation (0.1): In case a > 2 there are global solutions in time for a sufficiently small initial data (see FUJITA [6] and the survey paper of LEVINE [10]); in case 0 < a ~< 2 the non-trivial solution always blows up at some finite time.…”

Section: L'(t) = -U~(t L(t))mentioning

confidence: 99%

Global existence of solutions to one-phase Stefan problems for semilinear parabolic equations

Aiki

Imai

1998

Annali di Matematica pura ed applicata

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“…Moreover, they obtained that all time-global solutions are bounded. For more references of blow-up problems with free boundary condition, please see [1].…”

Section: Introductionmentioning

confidence: 99%

Blow-up and asymptotic behavior of solutions for reaction–diffusion equations with free boundaries

Sun

2015

Journal of Mathematical Analysis and Applications

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We study the blow-up phenomena and the asymptotic behavior of time-global solutions for reaction-diffusion equations u t = u xx + au + u p (a ∈ R and p > 1), with free boundaries and initial data σφ(x). We give a sharp threshold value σ * = σ * (φ, a, p) ≥ 0 such that the solution blows up in finite time when σ > σ * , vanishes (i.e. u → 0 as t → ∞) when σ < σ * , and when σ = σ * , it converges as t → ∞ to 0 or to an evenly decreasing positive stationary solution, depending on whether a ≥ 0 or a < 0. Moreover, when blow-up happens, we show that the blowup set is compact in the occupying domain of initial data and the free boundaries keep bounded.

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Behavior of free boundaries of blow-up solutions to one-phase stefan problems

Cited by 9 publications

References 5 publications

Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem

Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem

Global existence of solutions to one-phase Stefan problems for semilinear parabolic equations

Blow-up and asymptotic behavior of solutions for reaction–diffusion equations with free boundaries

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