RT Journal Article
T1 Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data
A1 Rodríguez Bernal, Aníbal
AB The title of the paper says it all. The author considers a reaction-diffusion equation with nonlinear boundary conditions on a bounded domain Ω⊂R N  . The initial value u 0   is allowed to be a function in L r (Ω) , with 1<r<∞ , as well as a measure on Ω . The nonlinear reaction term f  and the nonlinearity g  that is part of the Neumann-type boundary condition are both assumed to be locally Lipschitz and of critical growth at infinity. The author refers to his previous works, joint with Arrieta, Carvalho, Oliva, Pereira and Tadjine, for background results leading to this work. The key assumption in the present study is a balance term between the nonlinearities f  and g , (3.1), (5.1) or (6.5) in the paper, which the author interprets as "a condition that reflects a competition among diffusion, reaction and boundary flux''. In the case of power-like nonlinearities, the author obtains sharper results. Now you can read the title of the paper again.
PB Elsevier
SN 0022-0396
YR 2002
FD 2002-05
LK https://hdl.handle.net/20.500.14352/57911
UL https://hdl.handle.net/20.500.14352/57911
LA eng
NO H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations 72 (1998) 201–269.H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (Schmeisser and Triebel, Eds.), Text sur Mathematik, Vol. 133, pp. 9–126, Teubner, 1993. J. Arrieta and A. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc. 352 (1999), 285–310. J. Arrieta, A. Carvalho, and A. Rodríguez-Bernal, Critical exponents at the boundary, C. R. Acad. Sc. Ser. I 327 (1998), 353–358. J. Arrieta, A. N. Carvalho, and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 165 (1999), 376–406. J. Arrieta, A. N. Carvalho, and A. Rodríguez-Bernal, Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds, Comm. Partial Differential Equations 25(1/2) (2000), 1–37. H. Brezis, Problèmes unilatéraux, J. Math. Pure Appl. 51 (1972), 1–168. H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in "Contributions to Nonlinear Functional Analysis" (E. Zarantonello, Ed.), pp. 101–165, 1971. H. Brezis, and T. Cazenáve, A nonlinear heat equation with singular initial data, J. Anal. Math. 68 (1996), 277–304. H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), 73–97. A. N. Carvalho, S. M. Oliva, A. L. Pereira, and A. Rodríguez-Bernal, Attractors for parabolic problems with nonlinear boundary conditions, J. Math. Anal. Appl. 207 (1997), 409–461. M. Chipot, M. Fila, and P. Quittner, Stationary solution, blow up and convergence to stationary solution for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenian. 60 (1991), 35–103. L. Evans, Regularity properties of the heat equation subject to nonlinear boundary constraints, Nonlinear Anal. 1 (1997), 593–602.  A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Englewood Cliffs, NJ, 1964. A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34(2) (1985), 425–447. J. K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys Monogr. 25 (1988).  O. A. Ladyzenskaya, V. A. Solonikov, and N. N. Uralseva, "Linear and Quasi-linear Equations of Parabolic Type," American Mathematical Society, Providence, RI, 1968. H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations 16 (1974), 319–334. J. López-Gomez, V. Márquez, and N. Wolanski, Blow-up results and localization of blow-up points for the heat equations with a nonlinear boundary conditions, J. Differential Equations 92(2) (1991), 384–401. A. Rodríguez-Bernal and A. Tajdine, Nonlinear balance for reaction diffusion equations under nonlinear boundary conditions: Dissipativity and blow-up, J. Differential Equations 169 (2001), 332–372. R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1988. W. Walter, On the existence and nonexistence in the large of solution of parabolic differential equations with a nonlinear boundary conditions, SIAM J. Math. Anal. 6 (1975), 85–90. M. Wang and Y. Wu, Global existence and blow-up problems for quasilinear parabolic equations with nonlinear boundary conditions, SIAM J. Math. Anal. 24 (1993), 1515–1521.
NO DGES (Spain)
DS Docta Complutense
RD 7 may 2024