## Publication: Some qualitative dynamics of nonlinear boundary conditions

Loading...

##### Full text at PDC

##### Publication Date

2002-11

##### Authors

##### Advisors (or tutors)

##### Editors

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

World Scientific Publishing

##### Abstract

In this paper we survey some recent results on the behavior of solutions of parabolic equations subjected to nonlinear boundary conditions. The results range from local existence and regularity of solutions, to global existence, dissipativeness and existence of attractors, and to blow-up in finite time. Some applications are given to some singular perturbation problems and to pattern formation from boundary flux.

##### Description

##### UCM subjects

##### Unesco subjects

##### Keywords

##### Citation

Amann, H. [1993] “Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems," in Schmeisser/Triebel: Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, Vol. 133, pp. 9-126.
Angenent, S., Mallet-Paret, J. & Peletier, L. A. [1987] “Stable transition layers in a semilinear boundary value problem," J. Di_. Eqs. 67, 212-242.
Arrieta, J., Carvalho, A. N. & Rodríguez-Bernal, A. [1999] “Parabolic problems with nonlinear boundary conditions and critical nonlinearities," J. Diff. Eqs. 165, 376-406.
Arrieta, J., Carvalho, A. & Rodríguez-Bernal, A. [2000a] “Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds," Commun. Partial Di_. Eqs. 25(1/2), 1-37.
Arrieta, J., Carvalho, A. & Rodríguez-Bernal, A. [2000b] “Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions," J. Diff. Eqs. 168, 33-59.
Arrieta, J. M., Consul, N. & Rodríguez-Bernal, A. [2002] “Pattern formation from boundary reaction," Differential Equations and Dynamical Systems in Honor of Waldyr Oliva, Lisbon, June 2000, Fields Institute Communications Series, Vol. 31, pp. 13-18.
Benssousan, A., Lions, J. L. & Papanicolau, G. [1978] Asymptotic Analysis for Periodic Structures (North Holland).
Carvalho, A. N., Oliva, S. M., Pereira, A. L. & Rodríguez-Bernal, A. [1997] “Attractors for parabolic problems with nonlinear boundary conditions," J. Math. Anal. Appl. 207, 409-461.
Casten, R. & Holland, C. [1978] “Instability results for reaction diffusion equations with Neumann boundary conditions," J. Diff. Eqs. 27, 266-273.
Chipot, M., Fila, M. & Quittner, P. [1991] “Stationary solution, blow-up and convergence to stationary solution for semilinear parabolic equations with nonlinear boundary conditions," Acta Math. U. Comenian. 60, 35-103.
Consul, N. [1996] “On equilibrium solutions of diffusion equations with nonlinear boundary conditions," Z. Angew. Math. Phys. 47, 194-209.
Consul, N. & Solá-Morales, J. [1999] “Stability of local minima and stable nonconstant equilibria," J. Diff. Eqs. 157, 61-81.
Friedman, A. & McLeod, B. [1985] “Blow-up of positive solutions of semilinear heat equations," Indiana U. Math. J. 34(2), 425-447.
Hale, J. K. [1988] Asymptotic Behavior of Dissipative System, Mathematical Survey and Monographs, Vol. 25 (AMS).
Henry, D. [1981] Geometric Theory of Semilinear Parabolic Problems, Lecture Notes in Mathematics, Vol. 840 (Springer-Verlag).
Lacey, A. A. [1983] “Mathematical analysis of thermal runaway for spatially inhomogeneous reactions," SIAM J. Appl. Math. 43(6), 1350-1366.
Lacey, A. A., Ockendon, J. R. & Sabina, J. [1998] “Multidimensional reaction diffusion equations with nonlinear boundary conditions," SIAM J. Appl. Math. 58, 1622-1647.
Levine, H. [1971] “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = −_u + F(u)," Arch. Rat. Mech. Anal. 51, 371-386.
Levine, H. A. & Payne, L. E. [1974] “Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time," J. Diff. Eqs. 16, 319-334.
Matano, H. [1979] “Asymptotic behavior and stability of solutions of semilinear diffusion equations," Publ. Res. Inst. Math. Sci. 15, 401-454.
Matano, H. & Mimura, M. [1983] “Pattern formation in competition-di_usion systems in nonconvex domains," Publ. Res. Inst. Math. Sci. 19, 1049-1079.
Oleinik, O., Shamaev, A. S. & Yosifian, G. A. [1992] Mathematical Problems in Elasticity and Homogenization (North Holland).
Quittner, P. [1993] “On global existence and stationary solutions for two calses of semilinear parabolic problems," Comment. Math. Univ. Carolinae 34(1), 105-124.
Rocha, C. [1988] “Examples of attractors in scalar reaction-diffusion equations," J. Di_. Eqs. 73, 178-195.
Rodríguez-Bernal, A. [1998] “Localized spatial homogenization and large diffusion," SIAM J. Math. Anal. 29, 1361-1380.
Rodríguez-Bernal, A. & Tajdine, A. [2001] “Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: Dissipativity and blow-up," J. Diff. Eqs. 169, 332-372.
Rodríguez-Bernal, A. [2002] “Attractors parabolic problems with nonlinear boundary conditions, critical exponents and singular initial data," J. Diff. Eqs. 181, 165-196.
Sanchez-Palencia, E. [1980] Non Homogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127.
Sánchez-Hubert, J. & Sánchez-Palencia, E. [1989] Vibration and Coupling of Continuous Systems. Asymptotic Methods (Springer-Verlag).
Walter, W. [1975] “On the existence and nonexistence in the large of solution of parabolic differential equations with a nonlinear boundary conditions," SIAM J. Math. Anal. 6(1), 85-90.