Geodesic
On the uniqueness of initial boundary value problems for a system of semilinear parabolic equations with nonlinear nonlocal boundary conditions
Trudy Instituta matematiki, Tome 25 (2017) no. 2, pp. 60-69 
A. I. Nikitin. On the uniqueness of initial boundary value problems for a system of semilinear parabolic equations with nonlinear nonlocal boundary conditions. Trudy Instituta matematiki, Tome 25 (2017) no. 2, pp. 60-69. http://geodesic.mathdoc.fr/item/TIMB_2017_25_2_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider initial boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal Neumann boundary conditions and nonnegative initial data. We prove uniqueness and uniqueness of solutions.

[1] Yin Y., “On nonlinear parabolic equations with nonlocal boundary condition”, Journal of Mathematical Analysis and Applications, 185 (1994), 161–174 | DOI | MR | Zbl

[2] Chen B., Mi Y., “Quasilinear parabolic system with nonlocal boundary condition”, Boundary value problems, 2011 (2011), 18 pp. | MR

[3] Cortazar C., Elgueta M., Rossi J. D., “Uniqueness and nonuniqueness for the porous medium equation with non linear boundary condition”, Differential Integral Equations, 16 (2003), 1215–1222 | MR | Zbl

[4] Cortazar C., Elgueta M., Rossi J. D., “Uniqueness and non-uniqueness for a system of heat equations with non-linear coupling at the boundary”, Nonlinear Analysis, 37 (1999), 257–267 | DOI | MR | Zbl

[5] Gladkov A., Nikitin A., “A reaction-diffusion system with nonlinear nonlocal boundary conditions”, Int. J. Partial Differential Equations, 2014 (2014), 1–10 | DOI

[6] Gladkov A., Nikitin A., “On the existence of global solutions of a system of semilinear parabolic equations with nonlinear nonlocal boundary conditions”, Differential Equations, 52:4 (2016), 467–482 | DOI | MR | Zbl

[7] Zheng S., Kong I., “Roles of weight functions in a nonlinear nonlocal parabolic system”, Nonlinear Analysis: Theory, Methods and Applications, 68 (2008), 2406–2416 | DOI | MR | Zbl

[8] Gladkov A., Kim K. I., “Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition”, Journal of Mathematical Analysis and Applications, 338:1 (2008), 264–273 | DOI | MR | Zbl

[9] Gladkov A., Kim K. I., “Uniqueness and nonuniqueness for reaction-diffusion equation with nonlocal boundary condition”, Advances in Mathematical Sciences and Applications, 19:1 (2009), 39–49 | MR | Zbl

[10] Gladkov A., Guedda M., “Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition”, Nonlinear Analysis, 74 (2011), 4573–4580 | DOI | MR | Zbl

[11] Yang L., Fan C., “Global existence and blow-up of solutions to a degenerate parabolic system with nonlocal sources and nonlocal boundaries”, Monatshefte für Mathematik, 174:3 (2014), 493–510 | DOI | MR | Zbl

[12] Gladkov A., Kavitova T., “Blow-up problem for semilinear heat equation with nonlinear nonlocal boundary condition”, Applicable Analysis, 95 (2016), 1974–1988 | DOI | MR | Zbl

[13] Gladkov A., Kavitova T., “Initial boundary value problem for a semilinear parabolic equation with nonlinear nonlocal boundary conditions”, Ukrainian Mathematical Journal, 68:2 (2016), 162–174 | DOI | MR

[14] Marras M., Piro S. V., “Reaction-diffusion problems under non-local boundary conditions with blow-up solutions”, Journal of Inequalities and Applications, 2014:167 (2014), 11 pp. | MR

[15] Marras M., Piro S. V., “Explicit estimates for blow-up solutions to parabolic systems with time dependent coefficients”, Comtes Rendus de l'Academie Bulgare des Sciences, 67:4 (2014), 459–466 | MR | Zbl

[16] Nikitin A. I., “Lokalnoe suschestvovanie reshenii nachalno-kraevoi zadachi dlya sistemy polulineinykh parabolicheskikh uravnenii s nelineinymi nelokalnymi granichnymi usloviyami”, Vesnik Vitsebskaga dzyarzhaŭnaga ŭniversiteta, 2015, no. 5, 14–19

[17] Escobedo M., Herrero M. A., “A semilinear parabolic system in a bounded domain”, Annali di Matematica Pura de Applicata. Serie Quatra, 165 (1993), 315–336 | DOI | MR | Zbl

[18] Kahane C. S., “On the asymptotic behavior of solutions of parabolic equations”, Czechoslovac Mathematical Journal, 33(108) (1983), 262–285 | MR | Zbl

[19] Hu B., Yin H. M., “Critical exponents for a system of heat equations coupled in a non-linear boundary condition”, Mathematical Methods in the Applied Sciences, 19 (1996), 1099–1120 | 3.0.CO;2-J class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[20] Kartan A., Differentsialnoe ischislenie. Differentsialnye formy, Mir, M., 1971

[21] Cortazar C., del Pino M., Elgueta M., “On the short-time behavior of the free boundary of a porous medium eqution”, Duke Mathematical Journal, 87 (1997), 133–149 | DOI | MR | Zbl

[22] Aguirre J., Escobedo M., “A Cauchy problem for $u_t-\triangle u=u^p$ with $0

1.$ Asymptotic behaviour of solutions”, Annales de la Faculte des sciences de Toulouse: Mathematiques, 8:2 (1986–1987), 175–203 | DOI | MR | Zbl