Artículo

Arrieta, J.M.; Ferreira, R.; De Pablo, A.; Rossi, J.D. "Stability of the blow-up time and the blow-up set under perturbations" (2010) Discrete and Continuous Dynamical Systems. 26(1):43-61
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Abstract:

In this paper we prove a general result concerning continuity of the blow-up time and the blow-up set for an evolution problem under perturbations. This result is based on some convergence of the perturbations for times smaller than the blow-up time of the unperturbed problem together with uniform bounds on the blow-up rates of the perturbed problems. We also present several examples. Among them we consider changing the spacial domain in which the heat equation with a power source takes place. We consider rather general perturbations of the domain and show the continuity of the blow-up time. Moreover, we deal with perturbations on the initial condition and on parameters in the equation. Finally, we also present some continuity results for the blow-up set.

Registro:

Documento: Artículo
Título:Stability of the blow-up time and the blow-up set under perturbations
Autor:Arrieta, J.M.; Ferreira, R.; De Pablo, A.; Rossi, J.D.
Filiación:Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
Departamento de Matemática Aplicada, Universidad Carlos III de Madrid, 28911 Leganés, Spain
IMDEA Matemáticas, C-IX, Campus de Cantoblanco de la UAM, Madrid, Spain
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Palabras clave:Blow-up; Perturbations; Stability
Año:2010
Volumen:26
Número:1
Página de inicio:43
Página de fin:61
DOI: http://dx.doi.org/10.3934/dcds.2010.26.43
Título revista:Discrete and Continuous Dynamical Systems
Título revista abreviado:Discrete Contin. Dyn. Syst.
ISSN:10780947
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v26_n1_p43_Arrieta

Referencias:

  • Arendt, W., Approximation of degenerate semigroups (2001) Taiwanese J. Math., 5, pp. 279-295
  • Arrieta, J.M., Elliptic equations, principal eigenvalue and dependence on the domain, comm (1996) Partial Differential Equations, 21, pp. 971-991
  • Arrieta, J.M., Domain dependence of elliptic operators in divergence form, resenhas (1997) IMEUSP, 3, pp. 107-123
  • Arrieta, J.M., Spectral convergence and upper semicontinuity of attractors (2000) International Conference on Differential Equations (EQUADIFF'99), p. 615621. , Berlin. World Scientific
  • Arrieta, J.M., Carvalho, A.N., Spectral convergence and nonlinear dynamics of reaction difusión equations under perturbations of the domain (2004) J. Differential Equations, 199, pp. 143-178
  • Bändle, C., Brunner, H., Blow-up in diffusion equations (1998) A Survey, J. Comp. Appl. Math., 97, pp. 3-22
  • Baras, P., Cohen, L., Complete blow-up after Tmax for the solution of a semilinear heat equation (1987) J. Funct. Anal., 71, pp. 142-174
  • Chaves, M., Rossi, J.D., Regularity results for the blow-up time as a function of the initial data (2004) Differential Integral Equations, 17, pp. 1263-1271
  • Chen, X.Y., Matano, H., Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations (1989) J. Differential Equations, 78, pp. 160-190
  • Daners, D., Domain perturbation for linear and nonlinear parabolic equations (1996) J. Differential Equations, 129, pp. 358-402
  • Daners, D., Dirichlet problems on varying domains (2003) J. Differential Equations, 188, pp. 591-624
  • Daners, D., Perturbation of semi-linear evolution equations under weak assumptions at initial time (2005) J. Differential Equations, 210, pp. 352-382
  • Fermanian Kammerer, C., Merle, F., Zaag, H., Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view (2000) Math. Ann., 317, pp. 195-237
  • Friedman, A., McLeod, B., Blow-up of positive solutions of semilinear heat equations (1985) Indiana U. Math. J., 34, pp. 425-447
  • Galaktionov, V., Vazquez, J.L., Continuation of blow up solutions of nonlinear heat equations in several space dimensions (1997) Comm. Pure Appl. Math., 50, pp. 1-67
  • Galaktionov, V., Vazquez, J.L., The problem of blow-up in nonlinear parabolic equations (2002) Discrete Contin. Dyn. Syst., 8, pp. 399-433
  • Gidas, B., Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations (1981) Comm. Partial Differential Equations, 6, pp. 883-901
  • Giga, Y., A bound for global solutions of semilinear heat equations (1986) Commun. Math. Phys., 103, pp. 415-421
  • Groisman, P., Rossi, J.D., Dependence of the blow-up time with respect to parameters and numerical approximations for a parabolic problem (2004) Asymptot. Anal., 37, pp. 79-91
  • Groisman, P., Rossi, J.D., Zaag, H., On the dependence of the blow-up time with respect to the initial data in a semilinear parabolic problem (2003) Comm. Partial Differential Equations, 28, pp. 737-744
  • Henry, D., Geometric theory of semilinear parabolic equations (1981) Lecture Notes in Mathematics 840, , Springer-Verlag, Berlin
  • Herrero, M.A., Velazquez, J.J.L., Generic behaviour of one-dimensional blow up patterns (1992) Ann. Scuola Norm. Sup. di Pisa, 19, pp. 381-450
  • Merle, F., Solution of a nonlinear heat equation with arbitrarily given blow-up points (1992) Comm. Pure Appl. Math., 45, pp. 263-300
  • Quiros, F., Rossi, J.D., Vazquez, J.L., Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions (2002) Comm. Partial Differential Equations, 27, pp. 395-424
  • Quittner, P., Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems (2003) Houston J. Math., 29, pp. 757-799
  • Samarskii, A., Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P., Blow-up in quasilinear parabolic equations (1995) Walter de Gruyter, , Berlin

Citas:

---------- APA ----------
Arrieta, J.M., Ferreira, R., De Pablo, A. & Rossi, J.D. (2010) . Stability of the blow-up time and the blow-up set under perturbations. Discrete and Continuous Dynamical Systems, 26(1), 43-61.
http://dx.doi.org/10.3934/dcds.2010.26.43
---------- CHICAGO ----------
Arrieta, J.M., Ferreira, R., De Pablo, A., Rossi, J.D. "Stability of the blow-up time and the blow-up set under perturbations" . Discrete and Continuous Dynamical Systems 26, no. 1 (2010) : 43-61.
http://dx.doi.org/10.3934/dcds.2010.26.43
---------- MLA ----------
Arrieta, J.M., Ferreira, R., De Pablo, A., Rossi, J.D. "Stability of the blow-up time and the blow-up set under perturbations" . Discrete and Continuous Dynamical Systems, vol. 26, no. 1, 2010, pp. 43-61.
http://dx.doi.org/10.3934/dcds.2010.26.43
---------- VANCOUVER ----------
Arrieta, J.M., Ferreira, R., De Pablo, A., Rossi, J.D. Stability of the blow-up time and the blow-up set under perturbations. Discrete Contin. Dyn. Syst. 2010;26(1):43-61.
http://dx.doi.org/10.3934/dcds.2010.26.43