Universidad Nacional Mayor
de San Marcos
VI Simposio Internacional
de Matemática Aplicada
20-24 de Julio 2015
Acta de Resúmenes
Lima-Perú
Universidad Nacional Mayor de
San Marcos
Rector de la Universidad
Pedro Cotillo Zegarra
Vicerrectora Académica
Antonia Castro Rodrı́guez
Vicerrector de Investigación
Bernardino Ramı́rez Bautista
Decana de la Facultad
Doris Gómez Ticerán
Universidad Nacional Mayor de
San Marcos
Facultad de Ciencias Matemáticas
VI Simposio Internacional de
Matemática Aplicada
Comisión Organizadora
José Pérez Arteaga (Presidente)
Jaime Muñoz Rivera
Moisés Izaguirre Maguiña
Yolanda Santiago Ayala
Julio Flores Dionicio
Comité Cientı́fico
Jaime Muñoz Rivera, LNCC/RJ-Brasil
Verónica Poblete, U. Chile/Chile
Marcelo Cavalcanti, UEM/Paraná-Brasil
Octavio Vera Villagrán, UBB/Conc. Chile
Luis Carrillo Dı́az, UNMSM/Lima-Perú
Auspiciadores
Vicerrectorado Académico
Vicerrectorado de Investigación
Facultad de Ingenierı́a Industrial
Facultad de Quı́mica e Ingenierı́a Quı́mica
Facultad de Ingenierı́a Electrónica y Eléctrica
Facultad de Ing. Geológica, Minera, Metalúrgica y Geográfica
Presentación
La Facultad de Ciencias Matemáticas de la Universidad Nacional Mayor de
San Marcos, les expresa su más cálida bienvenida al VI Simposio de Matemática
Aplicada, el cual en esta oportunidad congrega a cientı́ficos del Brasil, Chile y
Perú. La realización de este Simposio, es fruto del esfuerzo de muchas voluntades, que confluyen en la participación de nuestros ilustres visitantes, quienes
junto a los conferencistas locales, darán realce a este magno evento.
Mención especial dedicamos al Profesor Dr. Jaime Edilberto Muñoz Rivera,
quien es el principal artı́fice de este evento, ya que haciendo gala de una profunda identificación con su alma mater, apoya constantemente las diversas
actividades de nuestra Facultad, ya sea propiciando la realización de estudios
de postgrado de nuestros alumnos destacados en importantes Centros de Investigación del exterior, ası́ como convocando periodicamente, en aulas sanmarquinas, a sus ilustres amistades del mundo cientı́fico. Por tal razón, la Comisión Organizadora le expresa su agradecimiento infinito y deseos de que su
proficua labor cientı́fica, continue por muchos años más. El Simposio alude a
la Matemática Aplicada, lo cual es una tendencia generalizada, en las diversas
investigaciones del mundo matemático, que sirven de sustento al desarrollo
tecnológico contemporáneo; ası́ observamos que los temas de las conferencias
y del Minicurso se ubican en dicha linea, abordando campos tan diversos como
Fisica, Quimica, Biologı́a, Economı́a, Ingenierı́a y muchas otras ramas afines.
Expresamos nuestro agradecimiento a los Conferencistas, ası́ como a las diversas instancias que hacen posible la realización de este VI Simposio de Matemática Aplicada.
La Comisión Organizadora
Conferencistas Visitantes
Jaime Muñoz Rivera1
UFRJ y LNCC-RJ-Brasil
Guiselda Alid Univ. Bio Bio-Chile
Xavier Carvajal UFRJ-RJ-Brasil
Mauricio Cataldo Uni. Bio Bio-Chile
Marcio Jorge Da Silva UE Londrina-Brasil
Lucie Harue Fatori UE Londrina-Brasi
Hugo Danilo Fernández Sare UFRJ-RJ-Brasil
Laura Senos Lacerda Fernández UF Juiz de Fora-Brasil
Ma To Fu USP-Brasil
Ricardo Fuentes Apolaya UFF-RJ-Brasil
Pedro Gamboa Romero UFRJ-RJ-Brasil
Erwan Hingant Univ. de Concepción-Chile
Adilandri Mércio Lobeiro Univ. Tecnologica de Paraná-Brasil
Verónica Poblete Univ. de Chile-Chile
Juan Carlos Pozo Vera Univ. Católica de Temuco-Chile
Julho Santiago Prates Filho UEM- Brasil
Amelie Rambaud Univ. Bio Bio-Chile
Carlos Alberto Raposo UF São João do Reis-MG-Brasil
Juan Amadeo Soriano Palomino UEM-Brasil
Octavio Vera Villagrán Univ. Bio-Bio-Chile
Victor Tapia Funes Univ. de Tarapacá-Chile
1 Prof.
Honorario de la UNMSM-Perú
Conferencistas locales
Renato Mario Benazic Tomé UNMSM
Eugenio Cabanillas Lapa UNMSM
Victor Rafael Cabanillas Zannini UNMSM
Luis Enrique Carrillo Dı́az UNMSM
Pedro Celino Espinoza Haro UNI
Roxana López Cruz UNMSM
José Raúl Luyo Sánchez UNMSM
Alfonso Pérez Salvatierra UNMSM
Mario Piscoya Hermoza2 UNMSM
Enrique Vásquez Huamán3 Universidad del Pacı́fico
Edgar Diógenes Vera Saravia UNMSM
2 Profesor
Emérito de la Universidad Nacional Mayor de San Marcos
Ex-Vice Rector Académico UNMSM
Ex-Decano FCM-UNMSM
3 Director de Desarrollo de la Universidad del Pacı́fico, profesor del Departamento
Académico de Economı́a y miembro del Centro de Investigación de esta casa de estudios.
VI SIMPOSIO INTERNACIONAL DE MATEMÁTICA APLICADA
20 al 24 de Julio de 2015
PROGRAMACIÓN
HORA
10:00-10:35
LUNES
10:35-11:35
11.35-12:35
12:35-13:10
MARTES
E. Vera
MIÉRCOLES
A. Pérez
M. da Silva
X. Carvajal
R. Benazic
A. Mércio
Ma To Fu
12:35-13:35
O. Vera
JUEVES
09:35-10:35
J. Soriano
C. Raposo
P. Gamboa
V. Tapia
VIERNES
A. Rodriguez
R. Fuentes
L. Lacerda
J. Luyo
CLAUSURA
15:00-15:35
15:35-16:35
16:35-17:35
17:35-18:35
18:35-19:35
INAUGURACIÓN
Conferencia
Inaugural
J. Muñoz
Intermédio
Musical
Artístico
Minicurso
L. Carrillo
L. Fatori
R. Cabanillas
E. Hingant
E. Cabanillas
J. Pozo
P. Espinoza
A. Rambaud
Minicurso
Minicurso
Minicurso
17:35 -18:10
R. López
18:10-19:10
V. Poblete
H. Fernández
17:35-18:10
E. Vásquez
M. Cataldo
18:10-19:35
MESA REDONDA
“La Universidad
en
Latinoamérica”
CENA
Conferencistas:
Locales □
Visitantes□
VI SIMPOSIO INTERNACIONAL DE MATEMÁTICA APLICADA
20 al 24 de Julio de 2015
Detalle de la Programación
Lunes 20
HORA
15:00-15:35
15:35-16:35
16:35-17:35
17:35-18:35
18:35-19:35
LUNES
INAUGURACIÓN
Conferencia
Inaugural
J. Muñoz
Intermédio
Musical
Artístico
Minicurso
L. Carrillo
L. Fatori
Lunes 20
15:00-15:35 Inauguración
Palabras del Presidente de la Comisión Organizadora
Palabras de la Decana de la Facultad de Ciencias Matemáticas
15:35-16:35 Conferencia Inaugural
“Estabilidad de sistemas dinámicos con múltiples mecanismos disipativos”
Dr. Jaime Muñoz Rivera.
16:35-17:35 Intermedio Musical Artístico
17:35-18:35 Minicurso
“Semigrupos aplicados a finanzas”
Dr. Luis Carrillo Díaz.
18:35-19:35 Conferencia
“The Lack of Exponential Stability to Boundary Dissipative Plates”
Dra. Luci Harue Fatori.
Martes 21, Miércoles 22 y Jueves 23
HORA
10:00-10:35
MARTES
E. Vera
MIÉRCOLES
A. Pérez
10:35-11:35
11.35-12:35
12:35-13:10
M. da Silva
X. Carvajal
R. Benazic
A. Mércio
Ma To Fu
12:35-13:35
O. Vera
15:00-15:35
15:35-16:35
16:35-17:35
R. Cabanillas
E. Hingant
E. Cabanillas
J. Pozo
P. Espinoza
A. Rambaud
Minicurso
17:35 -18:10
R. López
18:10-19:10
V. Poblete
Minicurso
H. Fernández
Minicurso
17:35-18:10
E. Vásquez
17:35-18:35
18:35-19:35
M. Cataldo
JUEVES
09:35-10:35
J. Soriano
C. Raposo
P. Gamboa
V. Tapia
18:10-19:35
MESA REDONDA
“La Universidad
en
Latinoamérica”
CENA
Martes 21
Mañana
10:00-10:35 “Por qué álgebra geométrica ?”
Dr. Edgar Vera Saravia.
10:35-11:35 “Taxas de decaimento para sistemas de Timoshenko não-homogêneos fracamente
dissipativos”
Dr. Marcio A. Jorge da Silva.
11:35-12:35 “Asymptotic behaviour of Solutions to a system of coupled Schrödinger equations”
Dr. Xavier Carvajal Paredes.
12:35-13:10 “Singularidades aisladas de foliaciones por curvas”
Dr. Renato Benazic Tomé.
Tarde
15:00-15:35 “TBA”
Dr. Rafael Cabanillas Zannini
15:35-16:35 “A natural slow-fast system arising in the scaling of the Becker-Döring equations”
Dr. Erwan Hingant.
16:35-17:35 Minicurso
“Semigrupos aplicados a finanzas”
Dr. Luis E. Carrillo Díaz.
17:35-18:10 “Bifurcaciones en un modelo depredador-presa tipo Leslie-Gower con retardo”
Dra. Roxana López Cruz.
18:10-19:10 “Fractional resolvent families of bounded semivariation”
Dra. Verónica Poblete.
Miércoles 22
Mañana
10:00-10:35 “Estudio del decaimiento exponencial para un problema de transmisión en
termoelasticidad unidimensional”
Dr. Alfonso Pérez Salvatierra.
10:35-11:35 “Solução das Equações de Saint Venant pelo Método das Características usando
Splines”
Dr. Adilandri Mércio Lobeiro.
11:35-12:35 “Energy decay of semilinear wave equations with moving boundary”
Dr. Ma To Fu.
12:35-13:35 “TBA”
Dr. Octavio Vera.
Tarde
15:00-15:35 “No-flux boundary problem involving p(x)-Laplacian-like operators via topological
methods”
Dr. Eugenio Cabanillas Lapa.
15:35-16:35 “Regularity of abstract Cauchy problem”
Dr. Juan Carlos Pozo Vera.
16:35-17:35 Minicurso
“Semigrupos aplicados a finanzas”
Dr. Luis Carrillo Díaz.
17:35-18:35 “Rates of decay for hyperbolic thermoelasticity”
Dr. Hugo D. Fernández Sare
18:35-19:35 “TBA”
Dr. Mauricio Cataldo.
Jueves 23
Mañana
09:35-10:35 “Exact controllability for Bresse system with variable coefficients”
Dr. Juan Amadeo Soriano Palomino.
10:35-11:35 “Existence and uniqueness of solution for a unilateral problem for the Klein-Gordon
Operator with Kirchhoff-Carrier nonlinearity”
Dr. Carlos Alberto Raposo.
11:35-12:35 “Exact solutions for long waves and blow-up phenomena”
Dr. Pedro Gamboa Romero.
12:35-13:10 “Estabilización de sistemas de control no lineal mediante el Principio de Reducción”
Mg. Víctor Tapia Funes.
Tarde
15:00-15:35 “Numerical irrelevant solutions (NIS) in nonlinear Elliptic eigenvalue problems”
Dr. Pedro C. Espinoza Haro.
15:35-16:35 “Stability in transmission problems to multicomponent Timoshenko beams with
localized Kelvin-Voigt dissipation”
Dra. Amelie Rambaud.
16:35-17:35 Minicurso
“Semigrupos aplicados a finanzas”
Dr. Luis Carrillo Díaz.
17:35-18:10 “TBA”
Dr. Enrique Vásquez.
18:10-19:35 MESA REDONDA
“La Universidad en Latinoamérica”.
Viernes 24
HORA
10:00-10:35
10:35-11:35
11.35-12:35
12:35-13:10
VIERNES
A. Rodriguez
R. Fuentes
L. Lacerda
J. Luyo
CLAUSURA
10:00-10:35 “Controlabilidad de cascaras de Naghdi con disipación localizada”
Mg. Alexis Rodriguez Carranza
10:35-11:35 “Global solutions and decay of a non linear coupled system with thermo-elastic”
Dr. Ricardo Fuentes Apolaya
11:35-12:35 “Riemann solutions for counterflow combustion in light porous foam”
Dra. Laura Senos Lacerda Fernández
12:35-13:10 “TBA”
Dr. José Luyo Sanchez
Resúmenes de las Conferencias
Índice
1. Semigrupos de Operadores Aplicados a Finanzas (Minicurso)
Luis Enrique Carrillo Dı́az
1
2. Estabilidad de sistemas dinámicos con múltiples mecanismos disipativos
Jaime Muñoz Rivera
3
3. Exact controllability for Bresse system with variable coefficients
Juan Amadeo Soriano Palomino y Rodrigo André Schultz
5
4. Exact controllability for Bresse system with variable coefficients
X. Carvajal, P. Gamboa, O. Vera
8
5. Asymptotic Behaviour of Solutions to a system of coupled Schrodinger equations
X. Carvajal
11
6. The Lack of Exponential Stability to Boundary Dissipative Plates
L.H. Fatori
12
7. Existence and uniqueness of solution for a unilateral problem for the
Klein-Gordon operator with Kirchhof-Carrier nonlinearity
C. Raposo, D. Perira, G. Araujo, A. Baena
14
8. Energy decay of semilinear wave equations with moving boundary
Ma To Fu
15
9. A natural slow-fast system arising in the scaling of the Becker-Döring
equations
Erwan Hingant
16
10. Taxas de decaimento para sistemas de Timoshenko não-homogêneos
fracamente dissipativos
17
11. Stability in Transmission Problems to Multicomponent Timoshenko
Beams With Localized Kelvin-Voigt Dissipation
J. Muñoz Rivera, A. Rambaud, O. Vera
18
12. Estudio del decaimiento exponencial para un problema de transmisión en termoelasticidad unidimensional
A. Pérez Salvatierra
19
13. Riemann Solutions For Counterflow Combustion in Light Porous Foam
L. Lacerda Fernández, G. Chapiro
20
14. Rates of Decay for Hyperbolic Thermoelasticity
Hugo Fernández Sare
21
15. No-Flux Boundary Problem Involving p(x)-Laplacian-Like Operators
via Topological Methods
Eugenio Cabanillas Lapa
22
16. Singularidades Aisladas de Foliaciones por Curvas
Renato Benazic Tomé
23
17. Numerical Irrelevant Solutions (NIS) in Nonlinear Elliptic Eingenvalue Problems
Pedro Espinoza Haro
24
18. Solução das Equações de Saint Venant pelo Método das Caracterı́sticas usando Splines
A. Mercio Lobeiro, M. Vieira Passos, J. Soriano Palomino
26
19. Fractional Resolvent Families of Bounded Semivariation
H. Henriquez, V. Poblete, J. Pozo
29
20. Estabilización de Sistemas de Control No Lineal Mediante el Principio de Reducción
Victor Tapia Funes
32
21. ¿Porqué Álgebra Geométrica?
Edgar Vera S.
33
22. Bifurcaciones en un modelo depredador-presa tipo Leslie-Gower con
retardo
Roxana López Cruz
34
23. Global solutions and decay of a non linear coupled system with thermoelastic
Ricardo Fuentes Apolaya
35
Semigrupos de Operadores Aplicados a Finanzas
[Minicurso]
Luis Enrique Carrillo Dı́az*
[email protected]
Resumen
Actualmente los especialistas en finanzas, demandan nuevos modelos matemáticos, que les permitan tomar decisiones con el menor riesgo posible. Desde
el año 1973, uno de los modelos más requeridos es el de Black-Scholes [2], quienes ganaron el Premio Nobel de economı́a en 1997. Estos modelos son empleados
en los Mercados de Opciones Financieras, lo cual está generando el desarrollo de
otras áreas de la propia matemática, en la intención de formular y hallar soluciones de nuevos modelos. A grosso modo, las opciones son contratos de compra o
venta a futuro, lo cual lleva implı́cita una cuota de incertidumbre, que se desea
minimizar sin detrimento de las ganancias. Como la dinámica de tales fenómenos
es gobernada por movimientos Brownianos, aparecen comportamientos que no
pueden ser representados con los métodos clásicos.
Este Minicurso, trata sobre un nuevo enfoque para Valoración de Opciones, el
cual hace uso de la Teorı́a de Semigrupos de Operadores y su herramienta principal,
el famoso teorema de Hille-Yosida. Se expone principalmente el protocolo para
valoración de Opciones europeas y americanas. Respecto a las opciones asiáticas
se establecen alternativas de solución en situaciones especiales.
Aprovechando la analogı́a entre Opciones Financieras y Opciones Reales, se aborda brevemente el problema de Opciones Reales americanas mediante semigrupo
de operadores.
Contenido
1. Conceptos fundamentales.
Breve introducción a Opciones: Opciones Financieras y Opciones Reales.
2. Teorı́a de Semigrupos: Teorema de Hille-Yosida.
3. Valoración de Opciones Financieras vı́a semigrupos de operadores.
• Valoración de Opciones europeas.
• Valoración de Opciones americanas.
• Valoración de Opciones asiáticas.
4. Valoración de Opciones Reales con semigrupos de operadores.
5. Aplicaciones.
* Profesor Principal de la Facultad de Ciencias Matemáticas
de la Universidad Nacional Mayor de San Marcos
1
Referencias
[1] Belleni-Morante, A. and McBride, A.C. Applied Nonlinear Semigroups; John Wiley
and Sons Ltda. (1998).
[2] Black, F. and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal
of Political Economy, (1973)
[3] Brandão-Cortazar, Evaluating Environmental Investments: A Real Options Approach PUC-DEI, Rio de Janeiro, Brasil, (2000).
[4] Colombo, F.,Giuli, M. and Vespri, V., A semigroup approach to no-arbitrage pricing
theory: constant elasticity variance model and term structure models, Progress in
Nonlinear Differential Equations and Their Appl. Vol 55, 113-126, Birkhäuser
(2003).
[5] Cruz-Baez, D.I., & González-Rodrı́guez, J.M., Semigroup Theory Applied to Options. Hindawi Publishing Corporation Journal of Applied Mathematics, 2-3
(2002) 131–139. Copyright (2002)
[6] Guimarães Dı́as, M. A., Análise de Investimentos com Opções Reais. Teorı́a e Prática
com Aplicações em Petróleo e em outros setores. Volume 1. Pre-Print, Brasil, Junho
(2013).
[7] Klaus-Engel, One-parameter semigroups for linear evolution equations; Springer,
New York, (1999).
[8] Kholodnyi, V. A. A Nonlinear Partial Differential Equation for American Options In
The Entire Domain Of The State Variable. Nonlinear Analysis, Theory Methods &
Application. v, Vol. 30, No. 8, pp. 5059-5070, (1997)
[9] Merton, R., Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, pp. 141–183. (1973).
[10] Muñoz Rivera, J., Estabilização de Semigrupos e Aplicações. Laboratorio Nacional
de Computação Cientifica. Petrópolis, Rio de Janeiro-Brasil (2009).
[11] Samanéz, C.P.; Ferreira, L. ; Do Nascimento, C. and Bisso, C., Evaluating the economy embedded in the Brazilian ethanol–gasoline flexfuel car: a Real Options approach,, Vol. 46, No. 14, 1565–1581,
http://dx.doi.org/10.1080/00036846.2013.877573
8
Applieds
Economics(2014).
[12] Sick, G. and Gamba, A., Some Important Issues Involving Real Options: An Overview, University of Calgary, (2005).
[13] Trigeorgis, L. Real Options: Managerial Flexibility and Strategy in Resource Allocation, MIT, Press, (1996)
2
Estabilidad de sistemas dinámicos con
múltiples mecanismos disipativos
Jaime E.Muñoz Rivera
Laboratorio Nacional de Computación Cientı́fica
Rua Getulio Vargas 333, CEP 25651-075, RJ Brasil
Instituto de Matemática-UFRJ
Resumen
En este trabajo estudiaremos los efectos de diversos mecanismos disipativos. Nuestra intension es mostrar, que la suma de estos mecanismos
no mejora en general la estabilización y puede suceder todo lo contrario,
esto es, que la estabilización sea muy lenta. En esta conferencia mostraremos como el orden de los mecanismos puede ocasionar diferentes tasas de
decaimiento. Adicionalmente mostraremos como se deben ordenar estos
mecanismos de tal forma de optimizar la tasa de decaimiento. Los mecanismos que consideraremos en esta exposición son: Mecanismo viscoso de
la clase de Kelvin-Voight, mecanismos tipo friccional, mecanismos térmicos. En los siguientes gráficos mostramos el caso de tres componentes, con
dos mecanismos disipativos.
3
Este trabajo fue realizado con la colaboración de Mauricio Sepulveda, Universidad de Concepción - Chile, Octavio Vera Villagrán, Universidad del Bio-Bio
- Chile y Margareth Alves, Universidad de Visoça - Brasil (UFV).
Referencias
[1] A. Borichev and Y. Tomilov: Optimal polynomial decay of functions and operator
semigroups. Mathematische Annalen. Vol. 347. 2455-478 (2009).
[2] K. Liu and Z. Liu: Exponential decay of the energy of the Euler Bernoulli beam with
locally distributed Kelvin-Voigt damping. SIAMJournal of Control and Optimization Vol. 36. 31086-1098 (1998).
[3] M. Alves, J.E. Muñoz Rivera, Mauricio Sepúlveda and O. Vera Villagrán: The
lack of exponential stability in certain transmission problems with localize kelvinvoigt dissipation. SIAM Journal of Applied Mathematics Volume 74, Número 2,
páginas 345 - 365 (2014).
[4] F. Ammar Khodja, A. Benabdallah, J.E. Muñoz Rivera and R. Racke: Energy
decay for Timoshenko systems of memory type, J. Differential Equations, 19482–115-(2003).
[5] Liu, Z., Rao, B.: Energy decay rate of the thermoelastic Bresse system, Z. Angew.
Math. Phys. Vol. 60, 54–69,(2009).
[6] Liu, Z., Rao, B.: Characterization of polynomial decay rate for the solution of linear
evolution equation, Z. Angew. Math. Phys. Vol. 56, pages 630–644, (2005).
[7] Liu, Z., Zheng, S.: Semigroups associated to dissipative systems, Chapman &
Hall/CRC Research Notes in Mathematics, 398 Vol. I (1999).
[8] Muñoz Rivera, J.E., Racke, R.: Global stability for damped Timoshenko systems,
Discr. Cont. Dyn. Sys. B, 9 1625–1639, (2003).
[9] Pruss, J.: On the spectrum of C0-semigroups, Trans. AMS 284, 847–857, (1984).
[10] Pruss, J., Batkai, A., K. Engel and Schnaubelt, R.: Polynomial stability of operator
semigroups, Math. Nachr. Vol. 279, (1), pages 1425-1440, (2006).
[11] Soufyane, A.: Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci. Paris, Sér.
I 328, 731–734, (1999).
[12] Timoshenko, S. P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, 6, 744-746;3, (1921).
4
Exact controllability for Bresse system with
variable coefficients
Juan Amadeo Soriano Palomino
Rodrigo Andre Schulz
Departamento de Matemática-DMA, UEM,
87020-900, Maringá, Avda Colombo, 5790, Campus Universitario
Maringá, PR, (44)30114040
E-mail: [email protected]
Abstract
This paper is concerned with the internal exact controllability of a generalized Bresse system with variable coefficients, which the controls functions acts in an arbitrarily small subinterval (l1 , l2 ) of (0, L). Our computation suggests a minimal time control and a region where the controls are
more effective. The variable coefficients can be viewed as a generalization
of Laplacian operator. The main result is obtained by applying Hilbert
Uniqueness Method proposed by Lions, without using the Holmgren’s
uniqueness theorem or the hypothesis of equal-speed waves of propagation.
Introduction
Consider the Bresse system given by


ρ ϕ − k(a(x)ϕx + ψ + lω)x − k0 l[ωx − lϕ] = h1 χ,


 1 tt
ρ2 ψtt − (b(x)ψx )x + k(ϕx + ψ + lω)
= h2 χ,



 ρ1 ωtt − k0 [c(x)ωx − lϕ]x + kl(ϕx + ψ + lω) = h3 χ,
(1)
in Q = (0, L) × (0, T ) where χ is the characteristic function of (L1 , l2 ) × (0, T ) and
(l1 , l2 ) ⊂ (0, L). Assume Dirichlet boundary conditions, that is,
ϕ(0, t) = ϕ(L, t) = ψ(0, t) = ψ(L, t) = ω(0, t) = ω(L, t),
for t ∈ (0, T ) and initial conditions given by


ϕ(., 0) = ϕ0 , ϕt (., 0) = ϕ1 ,



ψ(., 0) = ψ0 , ψt (., 0) = ψ1 ,



 ω(., 0) = ω0 , ωt (., 0) = ω1 .
(2)
(3)
The problem of exact controllability of (1) − (3) is formulated as follows:
Given T > 0, large enough, to find a Hilbert space H such that, for all initial
5
data {ϕ0 , ϕ1 , ψ0 , ψ1 , ω0 , ω1 } ∈ H, there are controls h1 = h1 (x, t), h2 = h2 (x, t) and
h3 = h3 (x, t), h1 , h2 , h3 ∈ L2 (l1 , l2 ) so that the solution {ϕ, ψ, ω} of (1) − (3) satisfies
ϕ(x, T ) = ϕt (x, T ) = ψ(x, T ) = ψt (x, T ) = ω(x, T ) = ωt (x, T ) = 0.
We apply the Hilbert Uniqueness Method (HUM) to obtain the exact controllability of (1) − (3).
1 Assumptions
In the observability and controllability results, we always take T > 2αR where
ρ1 ρ2 ρ1
α = max 1, , ,
,
k b0 k0
(4)
and
R = max l1 , L − l2 ,
(5)
2 Main result
Theorem 1
Assume that a, b, c ∈ W 1,∞ (0, L) satisfying a(x) ≥ 1, b(x) ≥ b0 > 0, and c(x) ≥
1 in (0, L). Let T > 2αR, α, and R given in (4) and (5), ϕ0 , ψ0 , ω0 ∈ H01 (0, L) and
ϕ1 , ψ1 , ω1 ∈ L2 (0, L). There are controls
h1 = h1 (x, t), h2 = h2 (x, t), h3 = h3 (x, t) ∈
L2 (0, T ; (l1 , l2 )) such that the solution ϕ, ψ, ω of the Bresse system

























ρ1 ϕtt − k(a(x)ϕx + ψ + lω)x − k0 l[ωx − lϕ] = h1 χ,
ρ2 ψtt − (b(x)ψx )x + k(ϕx + ψ + lω)
= h2 χ,
ρ1 ωtt − k0 [c(x)ωx − lϕ]x + kl(ϕx + ψ + lω) = h3 χ,
ϕ(0) = ϕ(L) = ψ(0) = ψ(L) = ω(0) = ω(L) = 0,
ϕ(x, 0) = ϕ0 (x), ϕt (x, 0) = ϕ1 (x),
ψ(x, 0) = ψ0 (x), ψt (x, 0) = ψ1 (x),
ω(x, 0) = ω0 (x), ωt (x, 0) = ω1 (x),
in (0, L) × (0T ),
in (0, L) × (0T )
in (0, L) × (0T )
in (0, L) × (0T )
x ∈ (0, L),
x ∈ (0, L),
x ∈ (0, L),
where χ is the characteristic function of interval (l1 , l2 ) checks
ϕ(x, T ) = 0,
ψ(x, T ) = 0,
ω(x, T ) = 0,
6
ϕ( x, T ) = 0,
ψ( x, T ) = 0,
ωt (x, T ) = 0.
(6)
References
[1] Liu Z, Rao B. Energy decay rate of the thermoelastic Bresse system.
Zeitschrift fur AngewandteMathematik und Physik 2009; 60:54-69.
[2] Soriano J.A, Muñoz Rivera J.E, Fatori L.H.: Bresse system with indefinite
damping. Journals of Mathematical Analysis and Applications; 387: 284290, (2011).
[3] Alabau Boussouira F, Muñoz Rivera J.E, Almeida Junior D.S.: Stability
to weak dissipative Bresse system. Journals of Mathematical Analysis and
Applications; 374(2), 481-498, (2011).
[4] Charles W, Soriano J.A, Falcão Nascimento F.A, Rodrigues J.H.: Decay
rates for Bresse system with arbitrary nonlinear localized damping. Journal
of Differential Equations; 8:2267-2290, (2013).
[5] Charles W., Soriano J.A, Schulz R.A.: Asymptotic stability for Bresse system. Journal of Mathematical Analysis and Applications; 412(1), 369-380,
(2014).
7
Exact controllability for Bresse system with
variable coefficients
X. Carvajal, P. Gamboa∗, & O. Vera†
Abstract
In this work we find exact solutions to the fifth-order KDV-BBM type
model that appear to describe the propagation of long waves in shallow
water. We study the possibility of blow-up phenomenon of the fifth-order
KDV-BBM type model under certain restrictions on the coeffcients. Moreover, by applying the Ince transformation we also establish exact travelling waves solutions to the nonlinear evolution equation Benney-Lin type.
1. Introduction
In this paper we will consider the initial value problem associated to the fifth
order BBM-KdV type equation
(
1
− 12
ηx2 − 41 η 3 = 0,
ηt + ηx − 61 ηxxt + δ1 ηxxxxt + δ2 ηxxxxx + 34 η 2 + γ η 2
xxx
x
x
x
η(x, 0) = η0 (x)
(1)
where η = η(x; t) is a real-valued function, and δ1 > 0, δ2 ; γ ∈ R. This model
was recently introduced by Bona et al [1] to describe the unidirectional propagation of water waves. It was formally obtained as a second order approximation from the higher order generalized Boussinesq system derived by Bona et
al [2], which describes the two-way propagation of water waves.
Finally we consider an equation of Benney-Lin type, that is,
ut + λ1 uxxxxx + λ2 uxxxx + uxxx + λ3 uxx + uux = 0
(2)
where x ∈ R; t > 0. u = u(x; t) is an unknown real-valued function. λ1 ; λ2 ; λ3 ∈
R are constant to be defined. When λ2 = λ3 , 0; the above equation is known
as Benney-Lin equation.
ut + λ1 uxxxxx + λ2 (uxxxx + uxx ) + uxxx + uux = 0; x ∈ R; t > 0;
∗ Instituto
(3)
de Matemática, Universidad Federal de Rio de Janeiro, Av. Athos da Silveira
Ramos, P.O. Box 68530, CEP:21945-970, RJ. Brazil E-mail address: [email protected] E-mail
address: [email protected]
† Departamento de Matemática, Universidad del Bı́o Bı́o, Collao 1202, Casilla 5C,Concepción. Chile E-mail address: [email protected]
8
where u = u(x; t) is an unknown real-valued function, λ1 , λ2 ∈ R and λ2 > 0. It
describes the propagation of one-dimensional small but finite amplitude long
waves in certain problems in fluids dynamics.
In this section we will prove that if γ = −1/30 and δ1 , δ2 satisfy the relation
√
9 388 1069 − 8269 δ1 25650δ2 = 190;
then an exact solution of (1) is
η(x; t) =
sec h2 (kx − ωt)
√
!2 + α;
3
1 − tan h(kx − ωt) −
sec h(kx − ωt)
2|k|
(4)
where Ck+ ;Ck− are constants that take values either 0 or 1, then,
lim η(x; t) = α −
x→±∞
4k 2 ±
C .
3 k
(5)
2. Blow-up phenomena
We start by recalling the concept of the blow-up solution. Let T be the maximal time of existence of the solution η(x; t). We say that the solution η has the
blow-up property in the space X if and only if
sup kη(t)kX = ∞.
t∈[0,T )
We say that the solution η does not have blow-up property in the space X
if
sup kη(t)kX < ∞.
t∈[0,T )
The solution in (4) have singularity along the line
s(t) =
where
ω
≈ 1.54978
k
−
ω
lnk0
t−
,
k
k
t ≥ 0,
(6)
lnk0
≈ 0.2492.
k
For the purpose of completing our paper we present the theorem
Theorem 3. Let T be the maximal time of existence of the solution η(x; t)
1
, then the corresponding solution blows-up
to the IVP (1). If δ1 > 0 and γ ≤
42
4
in H if and only if
lim inf
inf ηx (x, t) = −∞ or lim sup sup ηx (x, t) = ∞
−
t→T
x∈R
t→T −
(7)
x∈R
As our solution satisfies supx∈R |η(x; t)| = ∞, we concludes that η(x; t) have
blow-up in H 4 .
9
References
[1] Bona, J. L., Carvajal, X., Panthee, M., Scialom, M.: Higher-order models for
unidirectional water waves, preprint, 1-31.
[2] Bona J. L., Chen M. and Saut J.-C.: Boussinesq equations and other systems
for small-amplitude long waves in nonlinear dispersive media I. Derivation
and linear theory, J. Nonlinear Sci. 12, 283-318, (2002).
10
Asymptotic Behaviour of Solutions to a
system of coupled Schrödinger equations
Xavier Carvajal∗
Abstract
This paper is concerned with the behaviour of solutions to a system of
coupled Schrödinger equations


iut + ∆u + (α|u|2p + β|u|q |v|q+2 )u = 0,




ivt + ∆v + (α|v|2p + β|v|q |u|q+2 )v = 0,




u(x, 0) = ϕ(x),
v(x, 0) = ψ(x),
(1)
where x ∈ Rn , α, β ∈ R, p > 0 and q > 0. Which has applications in
many physical problems, especially in nonlinear optics. When the solution there exists globally we obtain the growth of the solutions in the
energy space. Also we find some conditions in order to obtain blow-up in
this space.
This work is jointly with Pedro Gamboa.
∗ Instituto
de Matemática, Universidad Federal de Rio de Janeiro, Av. Athos da Silveira
Ramos, P.O. Box 68530, CEP:21945-970, RJ. Brazil E-mail address: [email protected]
11
The Lack of Exponential Stability to
Boundary Dissipative Plates
L. H. Fatori ∗
Departamento de Matemática, Universidade Estadual de Londrina
86051-990 Londrina, PR, Brazil
J. E. Muñoz Rivera †
Laboratório de Nacional de Computação Cientı́fica, LNCC/MCT
25651-070 Petrópolis, RJ, Brazil
and
Instituto de Matemática, Universidade Federal do Rio de Janeiro
21945-970 Rio de Janeiro, RJ, Brazil
Abstract
In this work, we consider the plate equation with rotational term, with
dissipative mechanism effective in the interior of the domain and/or dissipative boundary condition. More specifically, let Ω be a bounded domain
of Rn types star-shape with smooth boundary ∂Ω = Γ0 ∪ Γ1 where Γ0 and
Γ1 are closed sets, disjoint and not empty of ∂Ω, as shown below.
and we consider the initial-boundary value problem
utt − γ∆utt + α∆2 u + a(x)ut = 0, in Ω × R+ ,
(1)
with boundary conditions
∂u
= 0, on Γ0 × R+
∂ν
∂u
∂∆u
∆u = 0,
−γ tt + α
= kut ,
∂ν
∂ν
(2)
u=
∗ Email:
† Email:
[email protected].
[email protected].
12
on
Γ1 × R+
(3)
and initial conditions
u(x, 0) = u0 (x),
ut (x, 0) = u1 (x) x ∈ Ω.
(4)
∂.
where γ , α and k are a positive constant, ∂ν
is the normal derivative with ν
an unit normal exterior vector to ∂Ω. We suppose that the function a ∈ L∞ (Ω)
and a(x) ≥ 0 a.e. Ω.
We study the asymptotic properties of the dissipative plate equation. Our
main result is that the system (1)-(4) does not decays exponentially to zero.
Our proof is based on the Weyl Theorem which means that the essential spectrum radius of an operator S is invariant by compact perturbations. Moreover,
from Borichev and Tomilov Theorem, we prove that the solution decays polynomially (slow) as t −1/6 as time goes to infinity.
References
[1] H. M. Berger,: A new approach to the analysis of large deflections of plates,
Journal of Applied Mechanics 22, 465-472, (1955).
[2] J. G. Eisley,: Nonlinear vibration of beams and rectangular plates, Z. Angew.
Math. Phys. 15, 167-175, (1964).
[3] J. H. Ginsberg,: Mechanical and Strutural Vibrations, Wiley, New York,
(2001).
[4] B. Z. Guo, W. Guo,: Adaptive stabilization for a Kirchhoff-type nonlinear
beam under boundary output feedback control, Nonlinear Anal. 66 427-441,
(2007).
[5] J. E. Lagnese:, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, vol. 10, SIAM, Pennsylvania, (1989).
[6] J. Lagnese, G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, J. Differential Equations 91, (1991), 355-388.
[7] J.L. Lions,: Quelques Méthodes de Résolution des Problèmes aux Limites Non
Linéaires, Dunod Gauthier-Villars, Paris, (1969).
[8] M. Reed, B. Simon,: Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, San Diego, (1978).
[9] S. Woinowsky-Krieger,: The effect of an axial force on the vibration of hinged
bars, J. Appl. Mech. 17 35-36, (1950).
[10] Borichev, A., Tomilov, Y.,: Optimal polynomial decay of functions and operator semigroups, Mathematische Annalen 347(2), 455–478, (2009).
13
Existence and uniqueness of solution for a
unilateral problem for the Klein-Gordon
operator with Kirchhof-Carrier nonlinearity
Carlos Raposo∗, Ducival Pereira†, Geraldo Araujo‡& Antonio Baena§
Abstract
This work deals with the unilateral problem for the operator of KleinGordon
∂2 u
L = 2 − M(|∇u|2 )∆u + M1 (|u|2 )u − f .
∂t
Using an appropriate penalization, see [1] and references therein, we obtain a variational inequality for the equation of Klein-Gordon perturbed
and then the existence and uniqueness of solutions is analyzed.
Acknowledgement. We would like to express our gratitude to the FAPEMIG
- Fundação de Amparo a Pesquisa do Estado de Minas Gerais.
References
[1] C. A. Raposo, D. C. Carvalho, G. M. Araujo and A. Baena. Unilateral Problems For the Klein-Gordon operator with nomliarity of Kirchhoff-Karrier
type. Electronic Journal of Differential Equations, Vol. 2015, pp. 1–14,
(2015).
∗ Department
of Mathematics, Federal University of São João del-Rei. São João del-Rei MG 36307-352, Brazil [email protected]
† Department of Mathematics, State University of Pará. Belém - PA 66113-200, Brazil [email protected]
‡ Department of Mathematics, Federal University of Pará Belém - PA 66075-110, Brazil
[email protected]
§ Department of Mathematics, Federal University of Pará Belém - PA 66075-110, Brazil
[email protected]
14
Energy decay of semilinear wave equations
with moving boundary
To Fu Ma
Instituto de Ciências Matemáticas e de Computação
Universidade de São Paulo
13566-590 São Carlos, SP, Brazil.
Abstract
This talk is dedicated to the energy stability of weakly damped semilinear wave equations defined on domains with moving boundary. Since
the boundary is a function of the time variable, the problem is intrinsically non-autonomous. Under the hypothesis that the lateral boundary
is time-like, the solution operator of the problem generates an evolution
process U (t, τ) : Xτ → Xt , where Xt are time-dependent Sobolev spaces.
Then, for non-contracting domains, we discuss the exponential stability
of the energy under time-dependent external forces.
References
[1] C. Bardos and G. Chen, Control and stabilization for the wave equation. III.
Domain with moving boundary, SIAM J. Control Optim. 19 (1981) 123-138.
[2] J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent
domain, J. Math. Anal. Appl. 42 (1973) 29-60.
[3] P. E. Kloeden, J. Real and C. Sun, Pullback attractors for a semilinear heat
equation on time-varying domains, J. Differential Equations 246 (2009)
4702-4730.
15
A natural slow-fast system arising in the
scaling of the Becker-Döring equations
Erwan Hingant
Ci2ma - Universidad de Concepción, Chile
joint work with Julien Deschamps and Romain Yvinec
Abstract
We will present the mathematical connection between two classical
models of phase transition phenomena describing different stages of cluster growth, namely, the Becker-Döring equations (BD) and the LifshitzSlyozov equation (LS). The former consist in an infinite set of ODE, one
for each size of clusters. While, the latter is a PDE on the density function
according to a continuous-size variable. Suitable scaling of BD with respect to a small parameter ε has been studied in [1, 4] where the authors
rigorously derive LS when the parameter ε → 0. In [2] we derive the same
limit in a stochastic context remarking that an underlying system on the
small size of clusters behave at a different time scale than the equation on
the density function. In the spirit of the works started by Fenichel in the
70’s, see e.g. [3], we take advantage of this sub-system to derive various
boundary conditions on the LS equation which were lacking in previous
works. Here, we focus on the deterministic version of this result.
References
[1] Jean-Francois Collet, Thierry Goudon, Frédéric Poupaud, and Alexis
Vasseur. The Beker-Döring system and its Lifshitz-Slyozov limit. SIAM
Journal on Applied Mathematics, 62(5):1488–1500 (electronic), (2002).
[2] Julien Deschamps, Erwan Hingant, and Romain Yvinec. From a stochastic Becker-Dr̈ing model to the Lifschitz-Slyozov equation with boundary
value. Preprint arXiv:1412.5025, (2014).
[3] Christian Kuehn. Multiple Time Scale Dynamics, volume 191 of Applied
Mathematical Sciences. Springer International Publishing, Cham, (2015).
[4] Philippe Laurençot and Stéphane Mischler. From the Becker-Döring
to the Lifshitz-Slyozov-Wagner equations. Journal of Statistical Physics,
106(5-6):957–991, (2002).
16
Taxas de decaimento para sistemas de
Timoshenko não-homogêneos fracamente
dissipativos
Marcio A. Jorge da Silva∗
[email protected]
Resumo
Nesta conferência serão abordados resultados sobre a existência e
taxas de decaimento para sistemas vigas de Timoshenko não homogêneos fracamente dissipativos. Neste caso, os coeficientes são funções
não constantes que podem variar de acordo com material que compõe
tais sistemas. Sendo assim, para a estabilização exponencial dos sistemas estudados, uma igualdade local das velocidades de propagação
de onda são consideradas como hipóteses. Quando tal condição local
não necessariamente vale, então decaimento polinomial é mostrado
para os sistemas de Timoshenko em geral.
Referências
[1] F. Ammar-Khodja, S. Kerbal and A. Soufyane, Stabilization of the nonuniform Timoshenko beam, J. Math. Anal. Appl. 327, 525-538,(2007).
[2] J. E. Muñoz Rivera and A. I. Ávila, Rates of decay to non homogeneous
Timoshenko model with tip body, J. Differential Equations 258 , no. 10,
3468-3490, (2015).
[3] A. Soufyane, Exponential stability of the linearized nonuniform Timoshenko beam, Nonlinear Anal. Real World Appl. 10, 1016-1020,
(2009).
[4] S. P. Timoshenko, Vibration Problems in Engineering, Van Nostrand,
New York, (1955).
∗ Universidade
Estadual de Londrina, Brasil
17
Stability in Transmission Problems to
Multicomponent Timoshenko Beams With
Localized Kelvin-Voigt Dissipation
J. Muñoz-Rivera∗& A. Rambaud†& O. Vera‡
[email protected]; [email protected] ; [email protected]
Abstract
We consider the transmission problem of Timoshenko’s beam composed of N components, each of them being either purely elastic (E), or a
Kelvin-Voigt viscoelastic material (V), or another elastic material inserted
with a frictional damping mechanism (F). Such material is illustrated in
Figure 1 for N = 7. We prove that the transmission problem is always wellposed. Our main result is that the rate of decay depends on the position of
each component. More precisely, we prove that the beam is exponentially
stable if and only if all the elastic components have at least one frictional
neighbouring material. Otherwise, the decay is only polynomial of order
1/t 2 .
———————Work partially supported by Fondecyt Project 11130378 and GIMNAP,
Depto de Matemática, Universidad del Bio-Bio.
∗ LNCC,
Rio de Janeiro, Brasil
del Bio-Bio, Concepción
‡ Universidad del Bio-Bio, Concepción
† Universidad
18
Estudio del decaimiento exponencial para
un problema de transmisión en
termoelasticidad unidimensional
Alfonso Pérez Salvatierra
UNMSM [email protected]
Abstract
En el presente trabajo estudiamos la existencia, unicidad de solución y
el decaimiento exponencial de la energı́a asociada al sistema de un problema de transmisión en termoelasticidad unidimensional representada
por,


u − αuxx + mθx + f (u) = h1 , en (L1 , L2 ) × (0, ∞)


 tt
(1)
θ
en (L1 , L2 ) × (0, ∞)

t − kθxx + muxt = h2 ,


 v − bv = h , en (0, L ) × (0, ∞)
tt
xx
3
1
Condiciones de frontera:


u(0, t) = θ(0, t) = v(L2 , t) = 0



u(L1 , t) = v(L1, t), αux (L1 , t) − mθ(L1 , t) = bvx (L1 , t)



 θ (L , t) = 0
x
(2)
1
Condiciones iniciales:


u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ (L1 , L2 )



θ(x,
0) = θ0 (x), x ∈ (L1 , L2 )



 v(x, 0) = v (x), v (x, 0) = v (x), x ∈ (0, L )
0
t
1
1
(3)
El cuerpo está compuesto por dos partes, una parte elástica y la otra parte
termoelástica. La prueba de la existencia y unicidad de la solución al sistema
se garantiza por el teorema de Lummer-Phillips y la estabilidad exponencial
por criterios del resolvente del operador matriz A(ρ(A).
References
[1] M. Alves, J.E. Muñoz Rivera, Mauricio Sepúlveda and O. Vera Villagrán:
The lack of exponential stability in certain transmission problems with localize
kelvinvoigt dissipation. SIAM Journal of Applied Mathematics Volume 74,
Número 2, páginas 345 - 365 (2014).
19
Riemann Solutions For Counterflow
Combustion in Light Porous Foam
Laura Senos Lacerda Fernández∗ &
Gigori Chapiro†
Abstract
In this talk we will consider a system of three evolutionary partial
diferential equations that models combustion of light porous foam under
air injection. We will see how to reduce this problem to an EDO by a
change of variables and we will study the existence of the combustion
wave sequences with negative velocity appearing in Riemann solutions.
References
[1] G. Chapiro, L. Senos; Riemann solutions for counter flow combustion in light
porous foam, preprint.
[2] G. Chapiro, D. Marchesin, and S. Schecter; Combustion waves and Riemann solutions in light porous foam, J. Hyper. Differential Equations,11,
295, (2014)
∗ Laura
Senos Lacerda Fernandez. Universidade Federal de Juiz de Fora, Juiz de Fora, MG
36036-900, Brazil Tel.: +55-32-2102-3308 Fax: +55-32-2102-3315 E-mail: [email protected]
† Grigori Chapiro. Universidade Federal de Juiz de Fora E-mail: [email protected]
20
Rates of Decay for Hyperbolic
Thermoelasticity
Hugo D.Fernández Sare∗
[email protected]
Abstract
We study models in thermoelasticity involving non-classical theory
for heat conduction. Results about stability of solutions for these systems
will be formulated.
∗ Instituto
de Matemática. Universidade Federal do Rio de Janeiro - Brasil
21
No-Flux Boundary Problem Involving
p(x)-Laplacian-Like Operators via
Topological Methods
Eugenio Cabanillas Lapa∗
[email protected]
Abstract
The purpose of this article is to obtain weak solutions for a class nonlinear elliptic problem for the p(x)-Laplacian-like operators under no-flux
boundary conditions. Our result is obtained using a Fredholm-type result
for a couple of nonlinear operators and the theory of the variable exponent Sobolev spaces.
∗ Universidad
Nacional Mayor de San Marcos-Lima-Perú
22
Singularidades Aisladas de Foliaciones por
Curvas
Renato Benazic Tomé*
[email protected]
Resumen
En esta conferencia hablaremos sobre puntos singulares aislados de un
sistema de n ecuaciones diferenciales complejas. Estudiaremos los principales invariantes y las formas normales cuando n = 2 y enumeramos
algunos resultados que pueden ser extendidos a dimensiones mayores a
2.
* Universidad
Nacional Mayor de San Marcos-Lima-Perú
23
Numerical Irrelevant Solutions (NIS) in
Nonlinear Elliptic Eingenvalue Problems
Pedro C. Espinoza Haro∗
Abstract
Consider the following the following kinds of nonlinear elliptic eigenvalue problems:
(
−∆u(x) = λf (u(x)) x ∈ Ω
(1)
u = 0, in ∂Ω
where Ω is a bounded open subset in Rn and whose border ∂Ω is smooth
and


a)f : [0, ∞) → R, es localmente Lipschitz continua



b)f has exactly 2m non-negatives zeros s0 = 0 < s1 < · · · < s2m−1
(2)



 and sig[f (t)] = (−1)i , ∀t ∈ (s , s ), i = 0, 1, . . . , 2m − 1
i
i+1
We will say that f in (2) satisfies the “positive area condition”
F (s2i+1 ) − F (s2i−1 ) > 0
(3)
Rt
for each i = 1, 2, . . . , m − 2, where F(t) = 0 f (s)ds.
The discrete analogue of (1) by Finite Difference whit suitable grid
points at is
(4)
Ax = λh2 f (x), x ∈ Rn
where A is a M-matrix, [9], [10], obtained in the discretization of operator
−∆, h is the mesh size and f (x) = (f (x1 ), . . . , f (xn )) is the Nemitskii operator
associated to scalar function f . In this note we study the numerically
irrelevant solutions (NIS) of the problem (1), which do not approximate
analytical solutions. This problem was studied, among others, by E. Bohl
[1] and W.J. Beyn & J. Lorenz [2]. Peitgen, Saupe and Schmitt, [6], [7],
[8], makes a detailed study applying techniques of topological degree and
theory of bifurcations. In this note we obtain, for the one-dimensional
case, some features of the NIS.
∗ Universidad
Nacional de Ingenierı́a-FIIS-Sección de Posgrado.
Ex-docente de la FCM-Universidad Nacional Mayor de San Marcos-Lima-Perú
24
References
[1] E. Bohl: On the bifurcation diagram of discrete analogues for ordinary bifurcation problems, Math. Methods Appl. Sci., v. 1, pp. 566-671, (1979).
[2] W.-J. Beyn & J. Lorenz: Spurious solutions for discrete superlinear boundary
value problems, Computing, v. 28, pp. 42-51 (1982).
[3] AK. J. Brown y H. Budin: On the existence of positive solutions for a class
of semi linear elliptic boundary value problems, SIAM J. Math Anal. Vol. 10,
Nº 5, 76-883,(1979)
[4] D.G. de Figueiredo: On the existence of multiple ordered solutions of nonlinear eigenvalue problems, Nonlinear Anal. Theory Meth. Appl., 11 pp.
481-492, (1987).
[5] P.C. Espinoza: Positive-ordered solutions of a discrete analogue of a nonlinear
elliptic eigenvalue problems, SIAM J. Numer. Anal. Vol. 31, N°3, 760-767,
(1994).
[6] H. O. Peitgen, D. Saupe y K. Schmitt: Nonlinear elliptic boundary problems
versus their finite aproximation: numerically irrelevant solutions, J. Reine
Angew Mathematik 322, 74-117, (1981).
[7] H. O. Peitgen y K. Schmitt: Positive and spurious solutions of Nonlinear eigenvalue problems, Springer Lectures Notes in Math. 878 275-324,
(1981).
[8] H. Jürgens, H.O. Peitgen and D. Saupe: Topological perturbations in the numerical study of nonlinear eigenvalue and bifurcation problems, Proceedings
Symposium on Analysis and Computation of Fixed Points, S. M. Robinson (ed.), New York-London, (1979).
[9] J. Schröder: M-matrices and generalizations using and operator theory approach, SIAM Review 20 213-244, (1978).
[10] R.S. Varga: Matirk iterative Analysis, Engle wood Cliffs. New Jersey,
(1962).
25
Solução das Equações de Saint Venant pelo
Método das Características usando Splines
Adilandri Mércio Lobeiro∗
Marlon Vieira Passos†
Juan Amadeo Soriano Palomino‡
[email protected]
[email protected]
[email protected]
Resumo
O presente trabalho apresenta a solução numérica das equações de
Saint Venant conjugando o Método das Características com Interpolações
Cúbicas com splines naturais no lugar de Interpolações Lineares usualmente adotadas para encontrar a velocidade média e o perfil da onda em
instantes de tempo pre-fixados.
1 Introdução
Na Engenharia Hidráulica, as equações de Saint Venant são frequentemente
usadas em estudos de escoamento não permanente em canais. No caso particular de canais retangulares de grande largura, as equações são
+ u ∂h
+ ∂u
= 0
∂x
∂x
(1)
∂h
+ u ∂u
+ g ∂x
= g(S0x − Sfx ),
∂x
(2)
∂h
∂t
∂u
∂t
em que u(x, t) é a velocidade média do escoamento (m/s) na direção x; h(x, t) é
a profundidade de fluxo (m); x é a distáncia ao longo do canal (m); t é o tempo
(s); g é a força gravitacional por unidade de massa (m/s2 ), S0x é a declividade
longitudinal e Sfx é a declividade da resistencia hidráulica na direção x, dado
por Sfx = |u| u/C 2 h, onde C é a constante de Chézy.
∗ Departamento
de Matemática, DAMAT, UTFPR
87301-899, Via Rosalina Maria dos Santos, 1233, bairro Area Urbanizada
† Coordenação de Engenharia Civil, COECI, UTFPR
Campo Mourão, PR, Fone:(44) 35181400
‡ Departamento de Matemática-DMA, UEM
87020-900, Maringá, Avenida Colombo, 5790, Campus Universitário-Maringá,
PR(44)30114040
26
Deseja-se obter a solução numérica das equações de Saint Venant via Método das Caracterásticas, que é um método consagrado por transformar um
sistema de Equações Diferenciais Parciais (EDPs) em um sistema de Equações
Diferenciais Ordinárias (EDOs) [1]. Neste caso, as equações (1) e (2) foram
transformadas em
dx
= u + c,
(3)
dt
d
(u + 2c) = g(S0x − Sfx ),
(4)
dt
dx
= u − c,
(5)
dt
d
(u − 2c) = g(S0x − Sfx ).
(6)
dt
As direções em (3) e (5) são chamadas direções características (C + e C − ,
respectivamente). As quantidades conservadas J + e J − dadas pelas equaç˜es
(4) e (6) ao longo das curvas características, são as invariantes de Riemann.
Para um estudo de caso, considerou-se um canal retangular de 400
mde
comprimento, 5 m de altura, 1 m de largura, declividade S0x = −0.0016 e Sfx
=
LP
0.5 uL |uL |/C 2 hL + uP |uP |/C 2 hP , onde C = 100 m(1/2) /s. Inicialmente o canal estava cheio de água e a mesma encontrava-se parada, ou seja, a velocidade inicial era zero. Considerou-se a descarga a esquerda, Figura 1, dada pela função
vazão qP , definida por
qP : R+ −→ R


−0.1t
se t ≥ 0 e t < 60



−6 + 0.1(t − 60) se t ≥ 60 e t < 80
(7)
t 7−→ qP (t) = 



−4
se
t ≥ 80.
p
A celeridade c é dada por cP = ghp , onde g é a constante gravitacional
e hP é a altura. Deseja-se calcular a propagação da onda pelo método das
características, ou seja, encontrar a velocidade média e a altura da água em
qualquer ponto do canal no decorrer do tempo.
Ao aplicar o Método das Características nestas equações, é de praxe utilizar
a interpolação linear para encontrar a velocidade e a profundidade da onda em
pontos não conhecidos da malha construída para obter a solução numérica [2].
A interpolação linear consiste em unir um conjunto de pontos com uma série
de linhas retas. Uma desvantagem desta aproximação é que não há diferenciação nos extremos de cada intervalo [3]. Para sanar esta dificuldade utilizou-se
a interpolação com Spline Cúbico Natural.
27
2 Resultados principais
Ao utilizar o Spline Cúbico Natural, permitiu-se obter a profundidade e velocidade do escoamento em posições específicas ao longo do comprimento do
canal e em instantes de tempo pré-fixados, o que tornou possível estimar tais
valores em qualquer ponto do canal, por meio de uma função duas vezes continuamente diferenciável. Sua utilização também otimizou o código teórico
por, entre outros fatores, não haver a necessidade de um número grande de
subdivisões no intervalo de comprimento estudado, uma clara vantagem se
comparada com a Interpolação Linear, que é comumente utilizada.
Referências
[1] García-Navarro, P.; Brufau, P.; Burguete, J.; Murillo, J.: The Shallow Water Equations: An Example of Hyperbolic System. Monografías de La Real
Academia de Ciencias de Zaragoza, 31, 89-119, (2008).
[2] Lobeiro, A. M.: Solução das Equações de Saint Venant em uma e duas dimensões usando o Método das Características. Universidade Federal do Paraná,
(2012).
[3] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P.: Numerical
Recipes. The Art of Scientific Computing. New York: Cambridge University
Press, (2007).
28
Fractional Resolvent Families of Bounded
Semivariation
Hernán R. Henrı́quez∗, Verónica Poblete†& Juan C. Pozo‡
[email protected]
[email protected]
[email protected]
Abstract
In this work we establish the existence of α-resolvent families of bounded
semivariation for all 0 < α < 2. We show that theory of cosine operator families
of bounded semivariation is a singular case of the theory of α-resolvent families.
Furthermore, by using the α-resolvent families of bounded semivariation and appropriate conditions on the forcing function, we study the existence of strong
solutions of non-homogeneous fractional differential equations. We consider the
autonomous and the non-autonomous cases.
Let X be a Banach space and suppose that A(t) : D(A(t)) ⊆ X → X are closed linear
operators with domain D(A(t)) = D for all t ∈ [0; a]; a > 0. We consider the following
problem
 α

D u(t) = A(t)u(t) + f (t, u(t)), t ∈ [0, a],


 t
u(0)
= x,
(1)



 u ′ (0) = y.
where α ∈ (1; 2), and the fractional derivative Dtα is understood in the Caputo sense.
If A(t) = A for all t ∈ [0; a], the problem 1 is known in the literature by fractional
abstract Cauchy problem associated to A of order α. The existence of solutions of this
problem is strongly related with the concept of α-resolvent family {Sα (t)}t≥0 ; introduced by Pruss [5] and widely developed by Bazhlekova [2]. In fact, the fractional
differential equation 1 is well posed if and only if A is the infinitesimal generator of
an α-resolvent family {Sα (t)}t≥0 . For more information see [[5], Proposition 1.1].
In the autonomous case, we assume that A generates an α-resolvent family {Sα (t)}t≥0
and define the operators P α (t) by
Pα (t)z = (gα−1 ∗ Sα ) (t)z,
t ≥ 0, z ∈ X.
The problem (1) has been studied in [4]. Specifically, they have established the
following resul[[4], Theorem 3.5].
∗ Universidad de Santiago, USACH, Departamento de Matemática, Santiago-Chile.
Partially supported by FONDECYT 1130144 and DICYT-USACH.
† Universidad de Chile, Facultad de Ciencias, Santiago-Chile.
Partially supported by PAIFAC 2015.
‡ Universidad Católica de Temuco, Departamento de Matemáticas y Fı́sica, Temuco-Chile.
Partially supported by FONDECYT 3140103.
29
Lemma 0.1. Assume that A generates an α-resolvent family {Sα (t)}t≥0 and x; y ∈ D(A).
Let
u(t) = Sα (t)x + (g1 ∗ Sα )(t)y + (Pα ∗ f ) (t); 0 ≤ t ≤ a.
(2)
The following conditions are equivalent:
(i) The α-resolvent family {Sα (t)}t≥0 is a family of bounded semivariation on [0; a].
(ii) For all function f ∈ C([0; a]; X), the function uis a strong solution of problem (1).
This result can also be obtained from the theory developed by H. Thieme [6]. On the
other hand, it has been showed in [3] that for α = 2 the conditions (i) and (ii) of Lemma
0.1 are in turn equivalent to A be a bounded linear operator. In what follows we will
show that for 1 < α < 2 there are α-resolvent families of bounded semivariation generated by unbounded operators. The following result establishes the Banach spaces
where α-resolvent family of bounded semi-variation can be defined. It is a generalization of the Baillon’s theorem about maximal regularity, [1].
Lemma 0.2. If A : D(A) ⊆ X → X is a closed linear operator which generates an αresolvent family {Sα (t)}t≥0 of bounded semivariation on [0, a] for all a > 0, then the
operator A is a bounded operator or X contains a closed subspace isomorphic to c0
(space of sequences convergent to 0).
In the non-autonomous case, in equation (1), we consider A(t) = A + B(t); where A
generates an α-resolvent family {Sα (t)}t≥0 with bounded semivariation and B : [0; a] →
L([D(A)]; X) is a strongly continuous map. Let ∆ = {f (t; s) : 0 ≤ s ≤ t ≤ a}. We denote
U (t; s)x = u(t) for (t; s) ∈ ∆.
We obtain the following results.
Corollary 0.3. The operator U (t; s) and B(t)U (t; s) have unique extensions to X, denoted by U (t; s) and W (t; s), respectively, and U ; W : ∆ → L(X) are strongly continuous
operator valued maps. Moreover,
Zt
U (t; s)x = Sα (t − s)x +
Pα (t − ψ)B(ψ)U (ψ; s)xdψ; x ∈ D(A);
s
U (t; s)x = Sα (t − s)x +
Z
t
s
Pα (t − ψ)W (ψ; s)xdψ; x ∈ X;
for all 0 ≤ s ≤ t ≤ a.
Corollary 0.4. Under appropriate conditions, problem (1) has a unique strong solution given by
Zt
Z t
u(t) = U (t; 0)x +
U (ψ; 0)ydψ +
Q(t; s)f (s)ds;
0
where
Z
t
Q(t; s)z =
s
0
gα−1 (τ − s)U(t; τ)zdτ; z ∈ X; 0 ≤ s ≤ t ≤ a.
30
References
[1] J. B. Baillon, Caractére borné de certains générateurs de semi-groupes linéaires dans
les espaces de Banach, C. R. Acad. Sci. Paris 290 ,757-760,(1980).
[2] E. G. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Eindhoven
University of Technology, Eindhoven, . Dissertation,(2001).
[3] D. Chyan, S. Shaw, S. Piskarev, On maximal regularity and semivariation of cosine
operator functions, J. London Mathematical Society 59 (3), 1023-1032, (1999).
[4] F. Li, M. Li, On maximal regularity and semivariation of α-times resolvent families,
Advances in Pure Mathematics 3, 680-684,(2013).
[5] J. Pruss, Evolutionary Integral Equations and Applications, Monographs Math. 87,
Birkhauser Verlag, Basel, (1993).
[6] H. Thieme, Differentiability of convolutions, integrated semigroups of bounded semivariation, and the inhomogeneous Cauchy problem, J. Evol. Equ. 8 (2), 283-305,
(2008).
31
Estabilización de Sistemas de Control No
Lineal Mediante el Principio de Reducción
Vı́ctor Tapia Funes*
[email protected]
Resumen
En el presente trabajo se expone las condiciones para la estabilización
de sistemas de control no lineal, mediante el principio de reducción, en
donde la teoria de la variedad central no se puede aplicar.
* Universidad
de Tarapaca - Sede Iquique
32
¿Porqué Álgebra Geométrica?
Edgar Vera Saravia*
Resumen
El Álgebra Geométrica fué creada por Clifford entre los años 1873 y
1879 como una generalización de los Cuaterniones de Hamilton. En esta
charla comentaremos el aspecto unificador de esta estructura y su empleo
en la fundamentación matemática de la fı́sica y sus aplicaciones en otras
áreas.
* Departamento
de Matemática
Universidad Nacional Mayor de San Marcos
33
Bifurcaciones en un modelo
depredador-presa tipo Leslie-Gower con
retardo
Roxana López-Cruz
Universidad Nacional Mayor de San Marcos, Perú
[email protected]
Resumen
Este trabajo trata acerca del modelo depredador-presa tipo Leslie-Gower
modificado (1), teniendo en cuenta que la población de presas es afectada
por un efecto Allee débil, ası́ como por un retardo τ en el crecimiento de
la población presa.
!
!


x(t
−
τ)
dx


= r 1−
(x − m) − qy x



K

 dt
(1)
Xµ : 




y
dy


= s 1−
y

dt
nx
Usamos el retardo τ como un parámetro de bifurcación. Demostramos la
aparición de una bifurcación de Hopf, cuando el retraso discreto cruza
cierta magnitud crı́tica.
Referencias
[1] Pallav. Jyoti Pal, Tapan. Saha and M. Sen Malay Banerjee, A delayed
predator-prey model with strong Allee effect in prey population growth,
Non Linear Dynamics Vol 68, Issue 1-2, pp 23-42, (2012)
[2] A. D. Bazykin, Nonlinear Dynamics of interacting populations, World
Scientific, (1998)
[3] C. C¸ elik, Hopf bifurcation of a ratio-dependent predator-prey system
with time delay, Chaos, Solitons and Fractals 42,1474-1484,(2009)
[4] E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma, J. D. Flores,
Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Applied Mathematical Modelling
35, 366-381, (2011)
[5] Y. Kuang, Delay differential equations with applications in Populations
Dynamics, Academic Press, Inc.(1993)
[6] H. Smith, An Introduction to Delay Differential Equations with Sciences
Applications, Springer (2011)
34
Global solutions and decay of a non linear
coupled system with thermo-elastic
Ricardo Fuentes Apolaya
Universidade Federal Fluminense-Rio de Janeiro-Brasil
[email protected]
Abstract
In this present work, the author prove the existence of global solutions
and the decay of nonlinear wave equation with thermo-elastic coupling
give by the system of equation:
′′
u (x, t) − µ(t)∆u(x, t) +
′
n
X
∂θ
i=1
∂xi
(x, t) + F(u(x, t)) = 0, in Q = Ω × (0, ∞)
θ (x, t) − ∆θ(x, t) +
n
X
∂u ′
i=1
∂xi
(x, t) = 0 in Q
where u is displacement, θ is absolute temperature, ∆ denotes the
Laplace operator, µ is a positive real function of t, F : R → R is continous function such that s.F(s) ≥ 0; Ω is a smooth bounded open set in Rn
with boundary Γ.
The non linearity F(v) = |v|ρ v usually appears in relativistic quantum
mechanic (see Segal [6] o Schiff [5]). Lions [3] studied the wave equation with the same non linearity, i.e., |v|ρ v, in a smooth bounded open
domain Ω of Rn and proved existence and uniqueness of solution using
both Faedo-Galerkin’s and Compactnesss’ methods. In [1] investigated
the system coupling with F(v) = |v|ρ v . They established global existence
and strong and weak solutions by Faedo-Galerkin’s method using a basis
of the space H01 (Ω) ∩ H 2 (Ω).
Based in the theory developed in the papers [1] , [4] and [7] Strauss
approximations of F, we will prove that the system coupling has a unique
global weak solution.
References
[1] R. Fuentes, H. Clark and A. Feitosa, On a nonlinear coupled system with
internal damping, Electronic Journal of Differential Equations, Volume
2000, 64,1-17, (2000).
[2] H. Brezis and T. Cazenave, Non Linear Evolution Equations, Lecture Notes
at. Instituto de Matemática, UFRJ, Rio de Janeiro, RJ, Brasil, (1994).
35
[3] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non
linèares, Dunod-Gauthier Villars, Paris, First edition, (1969).
[4] M. Milla and L. A. Medeiros, Hidden regularity for Semilinear Hyperbolic
Partial Differential Equations, An. Fac. des Sciences de Tolouse, Volume IX,
01, 103-120, (1988).
[5] I. Schiff, Non linear meson theory of nuclear forces, I. Physic. Rev., 84, 19,(1951).
[6] I. Segal, The global Cauchy problem for a relativistic scalar field with power
interaction, Bull. Soc. Math., France, 91, 129-135, (1963).
[7] W. Strauss, On weak solutions of semilinear hyperbolic equations, An. Acad.
Bras. Ciências Volume 42, 04, 645-651, (1950).
36
Mapa de la Ciudad Universitaria de San Marcos