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Characterization of turing diffusion-driven instability on evolving domains
Blow-up phenomena in reaction-diffusion systems
1. | Dipartimento di Matematica e Informatica, Università di Cagliari, 09123, Italy, Italy |
References:
[1] |
J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinearevolution equations, Quart. J. Math. Oxford, 28 (1977), 473-486.
doi: 10.1093/qmath/28.4.473. |
[2] |
C. Bandle and H. Brunner, Blow-up in diffusion equations, A survey, J. Comput. Appl. Math., 97 (1998), 3-22. |
[3] |
M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow-up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math.Univ. Comenian. (N.S.), LX (1991), 35103. |
[4] |
A. A. Lacey, Diffusion models with blow-up, J.Comput. Appl. Math., 97 (1998), 39-49.
doi: 10.1016/S0377-0427(98)00105-8. |
[5] |
J. López-Gómez, V. Márquez and N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition, J. Diff. Equ., 92 (1991), 384-401.
doi: 10.1016/0022-0396(91)90056-F. |
[6] |
J. López-Gómez, V. Márquez and N. Wolanski, "Global Behaviour of Positive Solutions to a Semilinear Equation with a Nonlinear Flux Condition," IMA Preprint Series, 810 , University of Minnesota, May 1991. |
[7] |
H. Kielhöfer, Halbgruppen und semilineare Anfangs-randwert-probleme, Manuscripta Math. 12 (1974), 121-152. |
[8] |
M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Numer. Funct. Anal. Optim., (2010), 32 (2011), 453-468. |
[9] |
L. E. Payne, G. A. Philippin and S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition I, Z. Angew. Math. Phys., 61 (2010), 971-978.
doi: 10.1007/s00033-010-0071-6. |
[10] |
L. E. Payne, G. A. Philippin and S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition II, Nonlinear Analysis, 73 (2010), 971-978.
doi: 10.1016/j.na.2010.04.023. |
[11] |
L. E. Payne and P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math., 48 (2008), 193-202. |
[12] |
G. A. Philippin and V. Proytcheva, Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problems, Math.Meth. Appl. Sci., 29 (2006), 297-307.
doi: 10.1002/mma.679. |
[13] |
P. Quittner, On global existence and stationary solutions of two classes of semilinear parabolic equations, Comm.Math. Univ.Carolinae, 34 (1993), 105-124. |
[14] |
P. Quittner and P. Souplet, "Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts, Basel, 2007. |
[15] |
B. Straughan, "Explosive Instabilities in Mechanics," Springer, Berlin, 1998. |
[16] |
J. L. Vázquez, The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation, Rend. Mat. Acc. Lincei s. IX, 15 (2004), 281-300. |
[17] |
F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102.
doi: 10.1512/iumj.1980.29.29007. |
[18] |
F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.
doi: 10.1007/BF02761845. |
show all references
References:
[1] |
J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinearevolution equations, Quart. J. Math. Oxford, 28 (1977), 473-486.
doi: 10.1093/qmath/28.4.473. |
[2] |
C. Bandle and H. Brunner, Blow-up in diffusion equations, A survey, J. Comput. Appl. Math., 97 (1998), 3-22. |
[3] |
M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow-up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math.Univ. Comenian. (N.S.), LX (1991), 35103. |
[4] |
A. A. Lacey, Diffusion models with blow-up, J.Comput. Appl. Math., 97 (1998), 39-49.
doi: 10.1016/S0377-0427(98)00105-8. |
[5] |
J. López-Gómez, V. Márquez and N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition, J. Diff. Equ., 92 (1991), 384-401.
doi: 10.1016/0022-0396(91)90056-F. |
[6] |
J. López-Gómez, V. Márquez and N. Wolanski, "Global Behaviour of Positive Solutions to a Semilinear Equation with a Nonlinear Flux Condition," IMA Preprint Series, 810 , University of Minnesota, May 1991. |
[7] |
H. Kielhöfer, Halbgruppen und semilineare Anfangs-randwert-probleme, Manuscripta Math. 12 (1974), 121-152. |
[8] |
M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Numer. Funct. Anal. Optim., (2010), 32 (2011), 453-468. |
[9] |
L. E. Payne, G. A. Philippin and S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition I, Z. Angew. Math. Phys., 61 (2010), 971-978.
doi: 10.1007/s00033-010-0071-6. |
[10] |
L. E. Payne, G. A. Philippin and S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition II, Nonlinear Analysis, 73 (2010), 971-978.
doi: 10.1016/j.na.2010.04.023. |
[11] |
L. E. Payne and P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math., 48 (2008), 193-202. |
[12] |
G. A. Philippin and V. Proytcheva, Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problems, Math.Meth. Appl. Sci., 29 (2006), 297-307.
doi: 10.1002/mma.679. |
[13] |
P. Quittner, On global existence and stationary solutions of two classes of semilinear parabolic equations, Comm.Math. Univ.Carolinae, 34 (1993), 105-124. |
[14] |
P. Quittner and P. Souplet, "Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts, Basel, 2007. |
[15] |
B. Straughan, "Explosive Instabilities in Mechanics," Springer, Berlin, 1998. |
[16] |
J. L. Vázquez, The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation, Rend. Mat. Acc. Lincei s. IX, 15 (2004), 281-300. |
[17] |
F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102.
doi: 10.1512/iumj.1980.29.29007. |
[18] |
F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.
doi: 10.1007/BF02761845. |
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