Solutions of the heat equation in domains with singularities

Aref'ev, V. N.; Bagirov, L. A.

doi:10.1007/bf02310297

Cited by 12 publications

(10 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Mihaǐlov papers in 1961 and 1963 [66][67][68] treated the case, where the parabolic boundary of the domain Q 0 lies below the characteristic plane t = 0 and has a typical form of the corresponding "backward" paraboloid of the fundamental solution 4 . In [67,68], this shape is perturbed by factor that makes the paraboloid "sharper", i.e., becomes "more regular"; see further comments on that.…”

Section: Higher-order Parabolic Pdes: On Some Known Regularity Resultsmentioning

confidence: 99%

“…5 Here, Kondrat'ev dealt with the singularity issues 3 Here and later on, we indicate only the results that are related to our blow-up setting to be explained shortly; these deep papers contain other involved conclusions. 4 Q.v. more recent Mihaǐlov's research on existence of boundary values of poly-harmonic functions for domains with smooth boundary; see references in [69].…”

Section: Higher-order Parabolic Pdes: On Some Known Regularity Resultsmentioning

confidence: 99%

“…For instance, his 1934 paper [74] was not cited practically in no such papers, but the one [4]. There were also discrepancies in treating whether or not Petrovskii derived the integral Osgood-Dini-like conditions (surely, he did already in 1934).…”

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

confidence: 99%

See 2 more Smart Citations

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.

View full text Add to dashboard Cite

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.

show abstract

Section: Higher-order Parabolic Pdes: On Some Known Regularity Resultsmentioning

confidence: 99%

Section: Higher-order Parabolic Pdes: On Some Known Regularity Resultsmentioning

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.

View full text Add to dashboard Cite

show abstract

“…In [AB96], [AB98] the first boundary problem is studied for the heat equation in a bounded plane domain with cuspidal points at the boundary at which the tangent coincides with a characteristic t = c, where c is a constant. The paper [AT12] contributed to the study of the first boundary problem for the 1D heat equation in a bounded plane domain by evaluating the first term of the asymptotic of a solution at the characteristic point.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point

2015

View full text Add to dashboard Cite

Bibliografische Information der Deutschen NationalbibliothekDie Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.de abrufbar. Abstract. The Dirichlet problem for the heat equation in a bounded domain G ⊂ R n+1 is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character. Universitätsverlag

show abstract

“…Study of a parabolic problem in a conical domain, to appear in Mathematical Journal of Okayama University] and , the same problem has been studied both in symmetric and non‐symmetric conical domains of

R^{3}

. The solvability of this kind of problems in the case of one‐dimensional space variable has been investigated, for instance, in Aref'ev and Bagirov where results concerning the behavior of the solution of the heat equation in various singular domains of

R^{2}

were obtained. Further references on the analysis of parabolic problems in non‐cylindrical domains are as follows: Alkhutov , Degtyarev , Labbas et al and Sadallah .…”

Section: Introductionmentioning

confidence: 99%

Study of the heat equation in a symmetric conical type domain of ℝN+1

Kheloufı

Sadallah

2013

Math. Meth. Appl. Sci.

View full text Add to dashboard Cite

This paper is devoted to the analysis of the N‐space dimensional heat equation, subject to Cauchy–Dirichlet boundary conditions. The problem is set in a symmetric conical type domain. More precisely, we look for sufficient conditions on the lateral boundary of the domain, as weak as possible in order to obtain the maximal regularity of the solution in an anisotropic Hilbertian Sobolev space. For this purpose, the domain decomposition method is employed. Copyright © 2013 John Wiley & Sons, Ltd.

show abstract

Solutions of the heat equation in domains with singularities

Abstract: ABSTRACT. We study the asyanptotics of solutions to the Dirichlet problem for the heat equation in timedependent domains with singular points.

Cited by 12 publications

References 9 publications

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

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

Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point

Study of the heat equation in a symmetric conical type domain of ℝN+1

Contact Info

Product

Resources

About