Abstract
We treat the Dirichlet problem for the 1D heat equation in a bounded domain \( \mathrm{g} \subset \mathbb{R}^2 \).The boundary of g is assumed to be smooth and noncharacteristic except for two points where it has contact of degree less than 2 with lines orthogonal to the t-axis. At these points the boundary has cuspidal singularities which have to be treated with particular care. We prove that this problem fits into the framework of analysis on manifolds with singular points elaborated by V. Rabinovich et al. (2000). The results extend to general parabolic equations.
Mathematics Subject Classification (2010). Primary 35K35; Secondary 35G15, 58J35.
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References
S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach spaces, Communs. Pure Appl. Math. 16 (1963), 121–239.
M.S. Agranovich and M.I. Vishik, Elliptic problems with a parameter and parabolic problems of general form, Russ. Math. Surv. 19 (1964), No. 3, 53–157.
V.N. Aref’ev and L.A. Bagirov, Asymptotic behavior of solutions of the Dirichlet problem for a parabolic equation in domains with singularities, Mat. Zam. 59 (1996), No. 1, 12–23.
V.N. Aref’ev and L.A. Bagirov, On solutions of the heat equation in domains with singularities, Mat. Zam. 64 (1998), No. 2, 163–179.
Th. Buchholz and B.-W. Schulze, Anisotropic edge pseudodifferential operators with discrete asymptotics, Math. Nachr. 184 (1997), 73–125.
R. Courant and D. Hilbert, Methoden der Mathematischen Physik, B. I u. II, 3. Auflage, Springer-Verlag, Berlin et al., 1968.
M.A. Evgrafov, Structure of solutions of exponential growth for some operator equations, Trudy Mat. Inst. im. V.A. Steklova 60 (1961), 145–180.
G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Acc. Naz. Lincei Mem., Ser. 8 5 (1956), 1–30.
V.A. Galaktionov, On regularity of a boundary point for higher-order parabolic equations: Towards Petrovskii type criterion by blow-up approach, arXiv: 0901.3986vI [math.AP] 26 Jan. 2009, 50 pp.
M. Gevrey, Sur les équations partielles du type parabolique, J. math. pure appl. Paris 9 (1913), 305–471; 10 (1914), 105–148.
I. Gokhberg and E. Sigal, An operator generalisation of the logarithmic residue theorem and the theorem of Rouché, Math. USSR Sbornik 13 (1971), No. 4, 603–625.
S.D. Ivasishen, Estimates of Green function of the first boundary value problem for a second order parabolic equation in a non-cylindrical domain, Ukr. Mat. Zh. 21 (1969), No. 1, 15–27.
J.J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of second order, Comm. Pure and Appl. Math. 20 (1967), No. 4, 797–872.
V.A. Kondrat’ev, Boundary problems for parabolic equations in closed domains, Trans. Moscow Math. Soc. 15 (1966), 400–451.
Th. Krainer, On the inverse of parabolic boundary value problems for large times, preprint 12, Institut für Mathematik, Universität Potsdam, 2002, 67 S.
O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, V. 23, Amer. Math. Soc., Providence, Rhode Island, 1968.
E.E. Levi, Sull’equazione del calore, Annali di Mat. 14 (1907), 187–264.
J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, V. 1–3, Dunod, Paris, 1968–1970.
V.G. Maz’ya and V.A. Plamenevskii, On the asymptotic behavior of solutions of differential equations in Hilbert space, Math. USSR Izvestiya 6 (1972), no. 5, 1067– 1116.
V.P. Mikhaylov, On the Dirichlet problem for a parabolic equation. I, II, Mat. Sb. 61 (103) (1963), 40–64; 62 (104) (1963), 140–159.
O.A. Oleynik, On linear second order equations with non-negative characteristic form, Mat. Sb. 69 (1966), 111–140.
I.G. Petrovskii, Zur ersten Randwertaufgabe der Wärmeleitungsgleichung, Compositio Math. 1 (1935), 389–419.
A. Piriou, Problèmes aux limites généraux pour des opérateurs différentiels paraboliques dans un domaine borné, Ann. Inst. Fourier (Grenoble) 21 (1971), No. 1, 59–78.
V. Rabinovich, B.-W. Schulze and N. Tarkhanov, A calculus of boundary value problems in domains with non-Lipschitz singular points. Math. Nachr. 215 (2000), 115– 160.
I.B. Simonenko, A new general method for investigating linear operator equations of the singular integral operator type. I, II, Izv. Akad. Nauk SSSR, Ser. Mat. 29 (1965), 567–586; 757–782.
L.N. Slobodetskii, The generalized spaces of S.L. Sobolevand their application to boundary value problems for partial differential equations, Uch. Zapiski Leningr. Ped. Inst. im. A.I. Gertsena 197 (1958), 54–112.
V.A. Solonnikov, Boundary value problems for linear parabolic systems of differential equations in general form, Proc. Steklov Inst. Math. 83 (1965), 3–162.
A.N. Tikhonov and A.A. Samarskii, Equations of Mathematical Physics, 4th Edition, Nauka, Moscow, 1972.
N. Wiener, The Dirichlet problem, J. Math. and Phys. Mass. Inst. Tech. 3 (1924), 127–146.
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Dedicated to V. Rabinovich on the occasion of his 70 th birthday
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Antoniouk, A., Tarkhanov, N. (2013). The Dirichlet Problem for the Heat Equation in Domains with Cuspidal Points on the Boundary. In: Karlovich, Y., Rodino, L., Silbermann, B., Spitkovsky, I. (eds) Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Operator Theory: Advances and Applications, vol 228. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0537-7_1
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