No Local<i>L</i><sup>1</sup>Solution for a Nonlinear Heat Equation

Çelik, Canan; Zhou, Zhengfang

doi:10.1081/pde-120025486

Cited by 20 publications

(23 citation statements)

References 11 publications

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“…The case q = 1 is more delicate, and is well known to be significantly more challenging. As remarked above, Celik & Zhou [6] showed that for the canonical equation…”

Section: Introductionmentioning

confidence: 80%

“…For this particular nonlinearity, given q ∈ (1, ∞), the pioneering results of [23,24,25] along with those of Giga [10] and Brezis & Cazenave [3] identify a critical exponent p ⋆ = 1 + 2q/d such that (1) with f satisfying (2) is locally well posed in L q if and only if p ≤ p ⋆ ; for p > p ⋆ one can find initial data in L q for which there is no local solution. While for q > 1 the equation is well behaved when p = p ⋆ , for the case q = 1 Celik & Zhou [6] showed that for the critical exponent p ⋆ = 1 + 2/d there are L 1 initial data for which there is no solution (resolving a problem posed in [3]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

Laister

Robinson

Sierżęga

et al. 2016

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We consider the scalar semilinear heat equationis continuous and non-decreasing but need not be convex. We completely characterise those functions f for which the equation has a local solution bounded in L q (Ω) for all non-negative initial data u 0 ∈ L q (Ω), when Ω ⊂ R d is a bounded domain with Dirichlet boundary conditions. For q ∈ (1, ∞) this holds if and only if lim sup s→∞ s −(1+2q/d) f (s) < ∞; and for q = 1 if and only if ∞ 1 s −(1+2/d) F (s) ds < ∞, where F (s) = sup 1≤t≤s f (t)/t. This shows for the first time that the model nonlinearity f (u) = u 1+2q/d is truly the 'boundary case' when q ∈ (1, ∞), but that this is not true for q = 1.The same characterisation results hold for the equation posed on the whole space R d provided that in addition lim sup s→0 f (s)/s < ∞.

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“…The case q = 1 is more delicate, and is well known to be significantly more challenging. As remarked above, Celik & Zhou [6] showed that for the canonical equation…”

Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

Laister

Robinson

Sierżęga

et al. 2016

Ann. Inst. H. Poincaré Anal. Non Linéaire

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show abstract

“…Lastly, the end-point case q = q * = 1 differs from the critical regime for q * > 1 in that there are non-negative data for which no non-negative solution may be defined [3,Theorem 11] and [4].…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of semilinear heat equations in L1

Laister

Sierżęga

2020

Ann. Inst. H. Poincaré Anal. Non Linéaire

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The problem of obtaining necessary and sufficient conditions for local existence of nonnegative solutions in Lebesgue spaces for semilinear heat equations having monotonically increasing source term f has only recently been resolved (Laister et al. (2016)). There, for the more difficult case of initial data in L 1 , a necessary and sufficient integral condition on f emerged. Here, subject to this integral condition, we consider other fundamental properties of solutions with L 1 initial data of indefinite sign, namely: uniqueness, regularity, continuous dependence and comparison. We also establish sufficient conditions for the global-in-time continuation of solutions for small initial data in L 1 .

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“…See Table 1. For a detailed history about the existence, nonexistence and uniqueness of (1.1), see [3,Section 1].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A doubly critical semilinear heat equation in the $$L^1$$ space

Miyamoto

2020

J. Evol. Equ.

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We study the existence and nonexistence of a Cauchy problem of the semilinear heat equationHere, N ≥ 1, p = 1 + 2/N and φ ∈ L 1 (R N ) is a possibly sign-changing initial function. Since N (p − 1)/2 = 1, the L 1 space is scale critical and this problem is known as a doubly critical case. It is known that a solution does not necessarily exist for every. In this paper we construct a local-in-time mild solution in L 1 (R N ) for φ ∈ X q if q ≥ N/2. We show that, for each 0 ≤ q < N/2, there is a nonnegative initial function φ 0 ∈ X q such that the problem has no nonnegative solution, using a necessary condition given by Baras-Pierre [Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), . Since X q ⊂ X N/2 (q ≥ N/2), X N/2 becomes a sharp integrability condition. We also prove a uniqueness in a certain set of functions which guarantees the uniqueness of the solution constructed by our method.

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No LocalL¹Solution for a Nonlinear Heat Equation

Cited by 20 publications

References 11 publications

A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

Well-posedness of semilinear heat equations in L1

A doubly critical semilinear heat equation in the $$L^1$$ space

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No LocalL1Solution for a Nonlinear Heat Equation

Cited by 20 publications

References 11 publications

A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

Well-posedness of semilinear heat equations in L1

A doubly critical semilinear heat equation in the $$L^1$$ space

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About

No LocalL¹Solution for a Nonlinear Heat Equation