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

Фила, Марек; Souplet, Philippe

doi:10.4171/ifb/43

Cited by 37 publications

(4 citation statements)

References 6 publications

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“…Especially for global solution to superlinear problem (2) (the constant α > 0) with a = 0 and b = 1, Ghidouche, Souplet, Tarzia and Fila ([17, 19, 33]) have viewed more clearly and thoroughly, and achieved rich and satisfactory results on existence of global solution and long-time behaviors of global solution. It should be remarked here that long-time behaviors of global solution have been classified in [19], which are named by fast solution and slow solution respectively in [17]: Let u be a solution of problem (2) with T = ∞.…”

Section: Huiling LI Xiaoliu Wang and Xueyan Lumentioning

confidence: 99%

“…The main purpose of the present paper is to generalize these results to the case where m(x) is any positive function and µ 1 = µ 2 . Analogous to the above [16] and [17,19], we will find conditions on the variable exponent function m(x) and the initial datum u 0 (x) to ensure existence or nonexistence of global solution and/or blowup solution. In addition, we will also analyze long-time behavior of global solution, and find conditions for existence of fast solution and/or slow solution.…”

Section: Huiling LI Xiaoliu Wang and Xueyan Lumentioning

confidence: 99%

“…This section is devoted to investigating long-time behavior of global solution for problem (1). According to [17], we define fast solution and slow solution of problem (1) as follows.…”

Section: 2mentioning

confidence: 99%

“…As in [17], we can find a sequence {t n } with t n → ∞ as n → ∞ and a positive constant ε < 1 such that…”

Section: 2mentioning

confidence: 99%

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A nonlinear Stefan problem with variable exponent and different moving parameters

Li¹,

Wang²,

Lu³

2020

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we consider a nonlinear diffusion problem with variable exponent, accompanied by double free boundaries possessing different moving parameters, where the variable exponent function m(x) satisfies that m(x) − 1 can change its sign. Local existence and uniqueness of solution are established firstly, and then, some sufficient conditions are achieved for finite time blowup, and as well for global existence. Asymptotic behavior is further investigated for global solution, and existences of fast solution and slow solution are presented by making use of upper-sub solutions, energy and scaling arguments.

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Section: Huiling LI Xiaoliu Wang and Xueyan Lumentioning

confidence: 99%

Section: Huiling LI Xiaoliu Wang and Xueyan Lumentioning

confidence: 99%

See 2 more Smart Citations

A nonlinear Stefan problem with variable exponent and different moving parameters

Li¹,

Wang²,

Lu³

2020

Discrete &Amp; Continuous Dynamical Systems - B

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Numerical approximation of a concrete carbonation model: Study of the ‐law of propagation

Zurek

2019

Numerical Methods Partial

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In this paper, we are interested in the long time behavior of approximate solutions to a free boundary model which appears in the modeling of concrete carbonation [1]. In particular, we study the long time regime of the moving interface. The numerical solutions are obtained by an implicit in time and finite volume in space scheme. We show the existence of solutions to the scheme and, following [2‐3], we prove that the approximate free boundary increases in time following a t‐law. Finally, we supplement the study through numerical experiments.

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A semilinear parabolic system with a free boundary

Wang

Zhao

2015

Z. Angew. Math. Phys.

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This paper deals with a semilinear parabolic system with free boundary in one space dimension. We suppose that unknown functions $u$ and $v$ undergo nonlinear reactions $u^q$ and $v^p$, and exist initially in a interval $\{0\leq x\leq s(0)\}$, but expand to the right with spreading front $\{x=s(t)\}$, with $s(t)$ evolving according to the free boundary condition $s'(t)=-\mu (u_x+\rho v_x)$, where $p,\, q,\, \mu, \,\rho$ are given positive constants. The main purpose of this paper is to understand the existence, uniqueness, regularity and long time behavior of positive solution or maximal positive solution. Firstly, we prove that this problem has a unique positive solution $(u,v,s)$ defined in the maximal existence interval $[0,T_{\max})$ when $p,\,q\geq 1$, while it has a unique maximal positive solution $(u,v,s)$ defined in the maximal existence interval $[0,T_{\max})$ when $p<1$ or $q<1$. Moreover, $(u,v,s)$ and $T_{\max}$ have property that either (i) $T_{\max}=+\infty$, or (ii) $T_{\max}<+\infty$ and $$ \limsup_{T\nearrow T_{\max}}\|u,\,v\|_{L^{\infty}([0,T]\times[0,s(t)])}=+\infty.$$ Then we study the regularity of $(u,v)$ and $s$. At last, we discuss the global existence ($T_{\max}=+\infty$), finite time blow-up ($T_{\max}<+\infty$), and long time behavior of bounded global solution.Comment: 26 page

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

Cited by 37 publications

References 6 publications

A nonlinear Stefan problem with variable exponent and different moving parameters

A nonlinear Stefan problem with variable exponent and different moving parameters

Numerical approximation of a concrete carbonation model: Study of the ‐law of propagation

A semilinear parabolic system with a free boundary

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