Reaction–Diffusion in Irregular Domains

Abdulla, Ugur G.

doi:10.1006/jdeq.2000.3761

Cited by 20 publications

(33 citation statements)

References 15 publications

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“…These purely technical assumptions can be got rid of [1]; our blow-up spectralboundary layer approach is also assumptionless and rigorous for the heat equation, see Sect. 7.7.…”

Section: On Some Details Of Petrovskii's Analysis In 1934-1935 and Exmentioning

confidence: 99%

“…Petrovskii's integral criterion of the Dini-Osgood type given in (3.11) is true in the N -dimensional radial case with (see [1]- [3] for the recent updating)…”

Section: On Some Details Of Petrovskii's Analysis In 1934-1935 and Exmentioning

confidence: 99%

“…However, some difficult questions remain open even for the second-order parabolic equations with order-preserving semigroups. We refer to [1][2][3]34,56,70] as a guide to a full history and the extensions of these important results.…”

Section: On Some Details Of Petrovskii's Analysis In 1934-1935 and Exmentioning

confidence: 99%

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On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach

Galaktionov

2009

Nonlinear Differ. Equ. Appl.

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Abstract. The classic problem of regularity of boundary points for higherorder partial differential equations (PDEs) is concerned. For second-order elliptic and parabolic equations, this study was completed by Wiener's (J. Math. Phys. Mass. Inst. Tech. 3:127-146, 1924) and Petrovskii's (Math. Ann. 109:424-444, 1934) criteria, and was extended to more general equations including quasilinear ones. Since the 1960-1970s, the main success was achieved for 2mth-order elliptic PDEs; e.g., by Kondrat'ev and Maz'ya. However, the higher-order parabolic ones, with infinitely oscillatory kernels, were not studied in such details. As a basic model, explaining typical difficulties of regularity issues, the 1D bi-harmonic equation in a domain shrinking to the origin (0, 0) is concentrated upon:where R(t) > 0 is a smooth function on [−1, 0) and R(t) → 0 + as t → 0 − . The zero Dirichlet conditions on the lateral boundary of Q0 and bounded initial data are posed:The boundary point (0, 0) is then regular (in Wiener's sense) if u(0, 0 − ) = 0 for any data u0, and is irregular otherwise. The proposed asymptotic blow-up approach shows that:(i) for the backward fundamental parabolae with R(t) = l(−t) 1/4 , the regularity of its vertex (0, 0) depends on the constant l > 0: e.g., l = 4 is regular, while l = 5 is not; (ii) for R(t) = (−t) 1/4 ϕ(− ln(−t)) with ϕ(τ ) → +∞ as τ → +∞, regularity/irregularity of (0, 0) can be expressed in terms of an integral Petrovskii-like (Osgood-Dini) criterion. E.g., after a special "oscillatory cut-off" of the boundary, the functioñ , a = constant ∈ (0, 3·23 ), together with typical ideas of boundary layers and blow-up matching analysis. Extensions to 2mth-order poly-harmonic equations in R N and other PDEs are discussed, and a partial survey on regularity/irregularity issues is presented.

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“…These purely technical assumptions can be got rid of [1]; our blow-up spectralboundary layer approach is also assumptionless and rigorous for the heat equation, see Sect. 7.7.…”

Section: On Some Details Of Petrovskii's Analysis In 1934-1935 and Exmentioning

confidence: 99%

“…Petrovskii's integral criterion of the Dini-Osgood type given in (3.11) is true in the N -dimensional radial case with (see [1]- [3] for the recent updating)…”

Section: On Some Details Of Petrovskii's Analysis In 1934-1935 and Exmentioning

confidence: 99%

See 1 more Smart Citation

On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach

Galaktionov

2009

Nonlinear Differ. Equ. Appl.

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“…In many cases this may be a characteristic single point. It should be mentioned that in the one-dimensional case Dirichlet and Cauchy-Dirichlet problems for the reaction-diffusion equations in irregular domains were studied in papers by the author [11,12]. Primarily applying this theory a complete description of the evolution of interfaces were presented in other papers [13,14].…”

Section: Boundary Value Problemsmentioning

confidence: 99%

“…Since the uniqueness and comparison results of this paper significantly improve the one-dimensional results from [11,12], we describe the one-dimensional results separately in Section 3. We prove Theorems 2.2, 2.6, and 2.7 in Sections 4-6, respectively.…”

Section: Boundary Value Problemsmentioning

confidence: 99%

Reaction-Diffusion in Nonsmooth and Closed Domains

Abdulla

2007

Boundary Value Problems

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We investigate the Dirichlet problem for the parabolic equationin a nonsmooth and closed domain Ω ⊂ R N+1 , N ≥ 2, possibly formed with irregular surfaces and having a characteristic vertex point. Existence, boundary regularity, uniqueness, and comparison results are established. The main objective of the paper is to express the criteria for the well-posedness in terms of the local modulus of lower semicontinuity of the boundary manifold. The two key problems in that context are the boundary regularity of the weak solution and the question whether any weak solution is at the same time a viscosity solution.

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