Some results on blow up for semilinear parabolic problems
| dc.book.title | Degenerate diffusions | |
| dc.contributor.author | Herrero, Miguel A. | |
| dc.contributor.author | Velázquez, J.J. L. | |
| dc.contributor.editor | Ni, Wei-Ming | |
| dc.contributor.editor | Peletier, L. A. | |
| dc.contributor.editor | Vázquez, Juan Luis | |
| dc.date.accessioned | 2023-06-20T21:07:51Z | |
| dc.date.available | 2023-06-20T21:07:51Z | |
| dc.date.issued | 1993 | |
| dc.description | Proceedings of the IMA Workshop held at the University of Minnesota, Minneapolis, Minnesota, May 13–18, 1991 | |
| dc.description.abstract | The authors describe the asymptotic behavior of blow-up for the semilinear heat equation ut=uxx+f(u) in R×(0,T), with initial data u0(x)>0 in R, where f(u)=up, p>1, or f(u)=eu. A complete description of the types of blow-up patterns and of the corresponding blow-up final-time profiles is given. In the rescaled variables, both are governed by the structure of the Hermite polynomials H2m(y). The H2-behavior is shown to be stable and generic. The existence of H4-behavior is proved. A nontrivial blow-up pattern with a blow-up set of nonzero measure is constructed. Similar results for the absorption equation ut=uxx−up, 0<p<1, are discussed. | |
| dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/22689 | |
| dc.identifier.doi | 10.1007/978-1-4612-0885-3_7 | |
| dc.identifier.isbn | 0-387-94068-5 | |
| dc.identifier.officialurl | http://link.springer.com/chapter/10.1007%2F978-1-4612-0885-3_7 | |
| dc.identifier.relatedurl | http://www.springer.com | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/60763 | |
| dc.issue.number | 47 | |
| dc.language.iso | eng | |
| dc.page.final | 125 | |
| dc.page.initial | 105 | |
| dc.page.total | 228 | |
| dc.publication.place | New York | |
| dc.publisher | Springer | |
| dc.relation.ispartofseries | IMA Volumes in Mathematics and its Applications | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 517.956.4 | |
| dc.subject.keyword | Semilinear parabolic problems | |
| dc.subject.keyword | blow up | |
| dc.subject.keyword | asymptotic behaviour of solutions | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.title | Some results on blow up for semilinear parabolic problems | |
| dc.type | book part | |
| dcterms.references | S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew Math. 390 (1988), 79–96. S. Angenent And B. Fiedler, The dynamics of rotating waves in scalar reaction-diffusion equations, Trans. Amer. Math. Soc. 307 (1988), 545–568. D.G. Aronson And H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Math. 30 (1978), 33–76. H. Bellout, A criterion for blow-up of solutions to semilinear heat equations, SI AM J. Math. Anal. 18 (1987), 722–727. J. Bebernes And S. Bricher, Final time blow up profiles for semilinear parabolic equations via center manifold theory, SIAM J. Math Anal., to appear. J. Bebernes, A. Bressan And D. Eberly, A description of blow-up for the solid fuel ignition model, Indiana Univ. Math. J. 36 (1987), 131–136. J. Bebernes And D. Eberly, Mathematical problems from combustion theory, Applied Mathematical Sciences v. 83, Springer Verlag (1989). M. Berger AND R.V. KOHN, A rescaling algorithm for the numerical calculation of blowing up solutions, Comm. Pure Appl. Math. 41 (1988), 841–863. C. Bandle AND I. Stakgold, The formation of the dead core in a parabolic reaction-diffusion equation, Trans. Amer. Math. Soc. 286 (1984), 275–293. A. Bressan, On the asymptotic shape of blow-up, Indiana Univ. Math. J. to appear. A. Bressan, Stable blow-up patterns, J. Diff. Equations, to appear. L.A. Caffarelli AND A. Friedman, Blow-up of solutions of nonlinear heat equations, J. Math. Anal, and Appl. 129 (1988), 409–419. X.Y. Chen AND H. Matano, Convergence, asymptotic periodicity and finite blow-up in one-dimensional semilinear heat equations, J. Diff. Equations, 78 (1989), 160–190. X. Chen, H. Matano and M. Mimura, Finite-point extinction and continuity of interfaces in a semilinear diffusion equation with strong absorption, to appear. J. Dold, Analysis of the early stage of thermal runaway, Quart. J. Mech. appl. Math. 38 (1985), 361–387. L.C. Evans AND B.F. Knerr, Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities, III, J. Math. 23 (1979), 153–166. S. Filippas and J.S. Guo, Quenching profiles for one-dimensional semilinear heat equations, to appear. S. Filippas AND R.V. Kohn, Refined asymptotics for the blow-up of ut- Δu = up, Comm. Pure Appl. Math., to appear,. A. Filippas AND J.B. Mc Leod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425–447. H. Fujita, On the blowing-up of solutions of the Cauchy problem for ut = ▵u + Au 1+ α, J. Fac. Sci. Univ. of Tokyo, Section I, 13 (1966), 109–124. V.A. Galaktionov, M.A. Herrero AND J.J.L. Velázquez, The space structure near a blow-up point for semilinear heat equations: a formal approach, USSR Comp. Math, and Math. Physics, 31, 3 (1991), 399–411. V.A. Galaktionov, M.A. Herrero AND J.J.L. Velázquez, The structure of solutions near an extinction point in a semilinear heat equation with strong absorption: a formal approach, To appear in the volume Nonlinear diffusion equations and their equilibrium states. N.G. Lloyd et al. ed., Brikhäuser Verlag. V.A. Galaktionov AND S.A. Posashkov, The equation ut = uxx + u β. Localization, asymptotic behaviour of unbounded solutions, Preprint N. 97 M., Keldysh Inst. Appl. Math. (1985) (in russian). V.A. Galaktionov AND S.A. Posashkov, Application of new comparison theorems in the investigation of unbounded solutions of nonlinear parabolic equations, Diff. Urav. 22, 7 (1986), 1165–1173. Y. Giga AND R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297–319. Y. Giga AND R.V. Kohn, Characterizing blow-up using similarity variables, Indiana Univ. Math. J. 36, (1987), 1–40. Y. Giga AND R.V. Kohn, Nondegeneracy of blow-up for semilinear heat equations, Comm. Pure Appl. Math. XLII(198), 845–884. D. Henry, Geometric theory of semilinear parabolic equations, Springer Lecture Notes in Mathematics No. 840 (1981). M.A. Herrero AND J.J.L. Velázquez, Blow up behaviour of one dimensional semilinear parabolic equations, Ann. Inst. Henri Poincaré, to appear. M.A. Herrero AND J.J.L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations, Integral and Differential Equations, to appear. M.A. Herrero And J.J.L. Velázquez, Plane structures in thermal runaway, Israel J. Maths., to appear. M.A. Herrero AND J.J.L. Velázquez, Generic behaviour of one-dimensional blow-up patterns, Annali della Sc. Norm. Sup. de Pisa, to appear. M.A. Herrero And J.J.L. Velázquez, Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption, J. Math. Anal, and Appl. to appear. L.M. Hocking, K. Stewartson And J.T. Stuart, A nonlinear instability burst in plane parallel flow, J. Fluid Mech. 51 (1972), 705–735. A.S. Kalashnikov, The propagation of disturbances in problems of nonlinear heat conduction with strong absorption, USSR. Comp. Math, and Phys. 14 (1974), 70–85. H. Kawarada, On solutions of initial boundary values problems for RIMS Kyoto U. 10 (1975), 729– A.A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math. 43 (1983), 1350–1386. H.A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form ut =-Au + F(u), Arch. Rat. Mech. Anal. 51, 5 (1973), 371–386. H.A. Levine, Quenching, nonquenching and beyond quenching for solutions of some parabolic equations, Annali di Mat. Pura ed Appl. CLV (1989), 243–250. H. Matano, Nonincrease of the lap number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci, Univ. Tokyo, Sec. IA, 29 (1982), 401–441. F. Merle, On the existence of a blow-up solution of the nonlinear heat equation with arbitrarily given blow-up points, to appear. J.J.L. Velázquez, Local behaviour near blow-up points for semilinear parabolic equation, J. Diff. Equations, to appear. F.B. Weissler, Single point blow-up of semilinear initial value problems, J. Diff. Equations 55 (1984), 204–224. D.V. Widder, The heat equation, Academic Press, New-York (1975). | |
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