Publication: Localization and blow-up of thermal waves in nonlinear heat-conduction with peaking
Loading...
Full text at PDC
Publication Date
1988
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
Citations
Exportar
Abstract
The authors consider the initial-boundary value problem for the porous medium equation ut =(um)xx in (0,∞)×(0,T), where m>1, 0<T<∞, with initial and boundary conditions u(x,0)= u0(x)≥0 in (0,∞), sup u0<∞, u0 has compact support, u(0,t)=ψ(t) for t (0,T), u0 and ψ are given nonnegative continuous functions and ψ(t)is monotonic increasing. The behaviour of the solution u(x,t) and the free boundary ζ(t)=sup{x[0,∞) : u(x,t)>0}as t↑T under the hypothesis that ψ(t)↑∞ as t↑T is investigated. The effect of localization of the blowing-up boundary function when lim sup t↑T ζ(t)<∞ is investigated. It is established that localization occurs if and only if lim sup t↑T (∫ t 0 ψ m (s)ds)/ψ(t)<∞, and some estimates concerning the asymptotic behaviour of the solution near the singular point t=T and in the blow-up set Ω={x≥0: lim sup t↑T u(x,t)=∞} are given. Various estimates from above and below on the length ω=supΩ of the blow-up set are obtained. These theorems make more precise some previous results concerning the localization of the boundary blowing-up function which were given in the book by A. A. Samarskiĭ, the reviewer et al. [Peaking modes in problems for quasilinear parabolic equations(Russian), "Nauka'', Moscow, 1987].
Proofs of the theorems are based on comparison with some explicit solutions and on construction of different kinds of weak sub- and supersolutions. The authors use some special integral identities and estimates of the solution and its derivatives by means of the maximum principle. A special comparison theorem above blow-up sets for different boundary functions is proved.
Description
UCM subjects
Unesco subjects
Keywords
Citation
Caffarelli, L.A, Friedman, A.: Continuity of the density of a gas flow in a porous medium. Trans. Am. Math. Soc. 252, 99-113 (1979)
Galaktionov, V.A., Kurdyumov, S.P., Mikhaílov, A.P., Samarskii, A.A.: Asymptotic stage of regimes with peaking and effective heat localization in nonlinear heat-conduction problems. Differ. Equations 16, 743-750 (1980). Translation of: Differ. Uravn. 16, 1196-1204 (1980)
Galaktionov, V.A., Samarskii, A.A.: Methods of constructing approximate self-similar solutions of nonlinear heat equations. I. Math. USSR-Sb. 46, 291-321 (1983). Translation of: Mat. Sb. (N.S.) 118, 291-322, 431 (1982)
Gilding, B.H.: Stabilization of flows through porous media. SIAM J. Math. Anal. 10, 237-246 (1979)
Gilding, B.H.: On a cIass of similarity solutions of the porous media equation. III. 1. Math. Anal. Appl. 77, 381-402 (1980)
Gilding, B.H.: Improved theory for a nonlinear degenerate parabolic equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (to appear)
Gilding, B.H., Peletier, L.A.: On a class of similarity solutions of the porous media equation. J. Math. Anal. Appl. 55, 351-364 (1976)
Gilding, B.H., Peletier, L.A.: On a class of similarity solutions of the porous media equation. II. J. Math. Anal. Appl. 57, 522-538 (1977)
Kalashnikov, A.S.: The occurrence of singularities in solutions of the non-steady seepage equation. U.S.S.R. Comput. Math. and Math. Phys. 7, 269-275 (1967). Translation of: Zh. Vychisl. Mat. i Mat. Fiz. 7, 440-444 (1967)
Kurdyumov, S.P., Kurkina, E.S., Malinetskii, G.G., Samarskii, A.A.: Dissipative structures in an inhomogeneous nonlinear burning medium. Soviet Phys. Dokl. 25, 167-169 (1980). Translation of: Dokl. Akad. Nauk SSSR 251, 587-591 (1980)
Kurdyumov, S.P., Malínetskii, G.G., Poveshchenko, Yu.A., Popov, Yu.P., Samarskii, AA: Interaction of dissipative thermal structures in nonlinear media. Soviet Phys. Dokl. 25, 252-254 (1980). Translation of: Dokl. Akad. Nauk SSSR 251, 836-839 (1980)
Likht, M.K.: On the propagation of perturbations in problems connected with degenerate quasilinear parabolic equations. Differential Equations 2, 496-498 (1966). Translation of: Dilferentsial'nye Uravneniya 2, 953-957 (1966)
Oleinik, O.A., Kalashnikov, A.S., Chzhou Y.-L.: The Cauchy problem and boundary problems for equations of the type of non-stationary filtration (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 22, 667-704 (1958)
Peletier, LA: The porous media equation. In: Amann, H., Bazley, N., Kirchgassner, K. (eds.) Applications of nonlinear analysis in the physical sciences, pp. 229-241. Boston: Pitman Advanced Publishing Program 1981
Samarskii, A.A., Elenin, G.G., Zmitrenko, N.V., Kurdyumov, S.P., Mikhailov, A.P.: The burning of a nonlinear medium in the form of complex structures. Soviet Phys. Dokl. 22, 737-739 (1977). Translation of: Dokl. Akad. Nauk SSSR 237, 1330-1333 (1977)
Samarskii, A.A., Galaktionov, VA, Kurdyumov, S.P., Mikhailov, A.P.: Localization of diffusion processes in media with constant properties. Soviet Phys. Dokl. 24, 543-545 (1979). Translation of: Dok1. Akad. Nauk SSSR 247, 349-353 (1979)
Samarskii, A.A., Zmitrenko, N.V., Kurdyumov, S.P., Mikhailov, A.P.: Effect of metastable localization of heat in a medium with nonlinear thermal conductivity. Soviet Phys. Dokl. 20, 554-556 (1976). Translation of: Dokl. Akad. Nauk SSSR 223, 1344-1347 (1975)
Samarskii,A.A., Zmitrenko, N.V., Kurdyumov, S.P., Mikhailov, A.P.: Thermal structures and fundamental length in a medium with nonlinear heat conduction and volumetric heat sources. Soviet Phys. Dokl. 21, 141-143 (1976). Translation of: Dokl. Akad. Nauk SSSR 227, 321-324 (1976)
Sattinger, D.H.: On the total variation of solutions of parabolic equations. Math. Ann. 183, 78-92 (1969)
Stokes, A.N.: Intersections of solutions of nonlinear parabolic equations. J. Math. Anal. Appl. 60, 721-727 (1977)
Zel'dovich, Ya.B., Raizer, Yu.P.: Physics of shock waves and high-temperature hydrodynamic phenomena. Vol. 1: New York: Academic Press 1966. Vol. 11: New York: Academic Press 1967