247
Lemma 7
Let y* be the function given in Lemma 3. Define
Then v, = ° on (0, T) x (0 ~w) and ify(t,·: vel is the corresponding solution of (Pp) we
have
II y*(t,·) ~ y(t,· : vel IIC(IT)::; c V t E [0, T].
Proof of Theorem 7. Let Ve be the function defined in Lemma 6. Then using Theorem
10 and lemmas 4, 5 and 6 we have that
II y(T: Vel ~ Yd IIC(IT) < II y*(T,·) ~ y(T,' : vel IIC(IT) + II y*(T,·) ~ Yd Ile(IT)
< II y*(T, .) ~ y(T . . : V e ) IIC(IT) + II fj( T •. ) - Yd Ile(IT-w)
+ II y*(T.·) ~ Yd Ile(w) +c ::; 4c
and the conclusion holds. •
References.
Adams, R.A. [1980]: Sobolev spaces, Academic Press, New York.
Alt, H.W. and Luckhaus, S. [1983]: Quasi/inmr Elliptic - Parabolic D~fferential Eq1tations. Math.
Z. 183. pp. 311-341.
Antontsev, S.N. and Dfaz, J.1. [1989]: New results on localization of solutions of nonlinear elliptic and parabolic equations obtained by energy methods, Soviet A1ath. DoH .. 38, pp
.53.5-.539.
Arino, 0., Gautier, S. and Penot, J.P. [1984]: A Fixed Point Theorem for sequentially contin1WUS mappings with applications to ordinary differential equations. Funkcialaj Ekvacioj,
27, pp. 273-279.
Aubin, T. [1982]: Nonlinear Analysis on Manifolds. ]I![onge-Ampere Eqllations. SpringerVerlag.
Bandle. C.,Dfaz, G. and Diaz, J.I. [1994]: Solutions d'equations de reaction-diffusion nonlineaires
explosant au bord parabolique. C.R.Acad.Scimces. Paris, 318, Serie I, pp. 4.5.5-460.
Barbu, V. [1976]: Nonlinear semigT'Oups and differential equations in Banach spaces. Noordhooff International Puplishing.
Benilan, Ph. [1981]: Evolution Equations and Acretiue Operator. Lecture Notes, Univ. of Kentucky.
Benilan, Ph. [1972]: Equations d'evolution dans un espace de Banach quelconque et application8.
These, Orsay.
Bermejo, R. [1994]: Numerical solution to a two-dimensional diffusive climate model. En "Modelado de Sistemas en Oceanografia, Climatologfa y Ciencias Medio - ambientales: Aspectos
Matematicos y Numericos". A. Valle and C.Pares eds. (Grupo de Analisis Matematico
Aplicado de la Universidad de Malaga).
Lemma 7
Let y* be the function given in Lemma 3. Define
Then v, = ° on (0, T) x (0 ~w) and ify(t,·: vel is the corresponding solution of (Pp) we
have
II y*(t,·) ~ y(t,· : vel IIC(IT)::; c V t E [0, T].
Proof of Theorem 7. Let Ve be the function defined in Lemma 6. Then using Theorem
10 and lemmas 4, 5 and 6 we have that
II y(T: Vel ~ Yd IIC(IT) < II y*(T,·) ~ y(T,' : vel IIC(IT) + II y*(T,·) ~ Yd Ile(IT)
< II y*(T, .) ~ y(T . . : V e ) IIC(IT) + II fj( T •. ) - Yd Ile(IT-w)
+ II y*(T.·) ~ Yd Ile(w) +c ::; 4c
and the conclusion holds. •
References.
Adams, R.A. [1980]: Sobolev spaces, Academic Press, New York.
Alt, H.W. and Luckhaus, S. [1983]: Quasi/inmr Elliptic - Parabolic D~fferential Eq1tations. Math.
Z. 183. pp. 311-341.
Antontsev, S.N. and Dfaz, J.1. [1989]: New results on localization of solutions of nonlinear elliptic and parabolic equations obtained by energy methods, Soviet A1ath. DoH .. 38, pp
.53.5-.539.
Arino, 0., Gautier, S. and Penot, J.P. [1984]: A Fixed Point Theorem for sequentially contin1WUS mappings with applications to ordinary differential equations. Funkcialaj Ekvacioj,
27, pp. 273-279.
Aubin, T. [1982]: Nonlinear Analysis on Manifolds. ]I![onge-Ampere Eqllations. SpringerVerlag.
Bandle. C.,Dfaz, G. and Diaz, J.I. [1994]: Solutions d'equations de reaction-diffusion nonlineaires
explosant au bord parabolique. C.R.Acad.Scimces. Paris, 318, Serie I, pp. 4.5.5-460.
Barbu, V. [1976]: Nonlinear semigT'Oups and differential equations in Banach spaces. Noordhooff International Puplishing.
Benilan, Ph. [1981]: Evolution Equations and Acretiue Operator. Lecture Notes, Univ. of Kentucky.
Benilan, Ph. [1972]: Equations d'evolution dans un espace de Banach quelconque et application8.
These, Orsay.
Bermejo, R. [1994]: Numerical solution to a two-dimensional diffusive climate model. En "Modelado de Sistemas en Oceanografia, Climatologfa y Ciencias Medio - ambientales: Aspectos
Matematicos y Numericos". A. Valle and C.Pares eds. (Grupo de Analisis Matematico
Aplicado de la Universidad de Malaga).