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Universidad Nacional Mayorde San Marcos
VI Simposio Internacionalde Matematica Aplicada
20-24 de Julio 2015
Acta de Resúmenes
Lima-Peru
Universidad Nacional Mayor deSan Marcos
Rector de la UniversidadPedro Cotillo Zegarra
Vicerrectora AcademicaAntonia Castro Rodrıguez
Vicerrector de InvestigacionBernardino Ramırez Bautista
Decana de la FacultadDoris Gomez Ticeran
Universidad Nacional Mayor deSan Marcos
Facultad de Ciencias Matematicas
VI Simposio Internacional deMatematica Aplicada
Comision OrganizadoraJose Perez Arteaga (Presidente)Jaime Munoz RiveraMoises Izaguirre MaguinaYolanda Santiago AyalaJulio Flores Dionicio
Comite CientıficoJaime Munoz Rivera, LNCC/RJ-Brasil
Veronica Poblete, U. Chile/ChileMarcelo Cavalcanti, UEM/Parana-Brasil
Octavio Vera Villagran, UBB/Conc. ChileLuis Carrillo Dıaz, UNMSM/Lima-Peru
Auspiciadores
Vicerrectorado AcademicoVicerrectorado de InvestigacionFacultad de Ingenierıa Industrial
Facultad de Quımica e Ingenierıa QuımicaFacultad de Ingenierıa Electronica y Electrica
Facultad de Ing. Geologica, Minera, Metalurgica y Geografica
Presentacion
La Facultad de Ciencias Matematicas de la Universidad Nacional Mayor deSan Marcos, les expresa su mas calida bienvenida al VI Simposio de MatematicaAplicada, el cual en esta oportunidad congrega a cientıficos del Brasil, Chile yPeru. La realizacion de este Simposio, es fruto del esfuerzo de muchas volunta-des, que confluyen en la participacion de nuestros ilustres visitantes, quienesjunto a los conferencistas locales, daran realce a este magno evento.Mencion especial dedicamos al Profesor Dr. Jaime Edilberto Munoz Rivera,quien es el principal artıfice de este evento, ya que haciendo gala de una pro-funda identificacion con su alma mater, apoya constantemente las diversasactividades de nuestra Facultad, ya sea propiciando la realizacion de estudiosde postgrado de nuestros alumnos destacados en importantes Centros de In-vestigacion del exterior, ası como convocando periodicamente, en aulas san-marquinas, a sus ilustres amistades del mundo cientıfico. Por tal razon, la Co-mision Organizadora le expresa su agradecimiento infinito y deseos de que suproficua labor cientıfica, continue por muchos anos mas. El Simposio alude ala Matematica Aplicada, lo cual es una tendencia generalizada, en las diversasinvestigaciones del mundo matematico, que sirven de sustento al desarrollotecnologico contemporaneo; ası observamos que los temas de las conferenciasy del Minicurso se ubican en dicha linea, abordando campos tan diversos comoFisica, Quimica, Biologıa, Economıa, Ingenierıa y muchas otras ramas afines.Expresamos nuestro agradecimiento a los Conferencistas, ası como a las di-versas instancias que hacen posible la realizacion de este VI Simposio de Ma-tematica Aplicada.
La Comision Organizadora
Conferencistas Visitantes
Jaime Munoz Rivera1 UFRJ y LNCC-RJ-Brasil
Guiselda Alid Univ. Bio Bio-ChileXavier Carvajal UFRJ-RJ-BrasilMauricio Cataldo Uni. Bio Bio-ChileMarcio Jorge Da Silva UE Londrina-BrasilLucie Harue Fatori UE Londrina-BrasiHugo Danilo Fernandez Sare UFRJ-RJ-BrasilLaura Senos Lacerda Fernandez UF Juiz de Fora-BrasilMa To Fu USP-BrasilRicardo Fuentes Apolaya UFF-RJ-BrasilPedro Gamboa Romero UFRJ-RJ-BrasilErwan Hingant Univ. de Concepcion-ChileAdilandri Mercio Lobeiro Univ. Tecnologica de Parana-BrasilVeronica Poblete Univ. de Chile-ChileJuan Carlos Pozo Vera Univ. Catolica de Temuco-ChileJulho Santiago Prates Filho UEM- BrasilAmelie Rambaud Univ. Bio Bio-ChileCarlos Alberto Raposo UF Sao Joao do Reis-MG-BrasilJuan Amadeo Soriano Palomino UEM-BrasilOctavio Vera Villagran Univ. Bio-Bio-ChileVictor Tapia Funes Univ. de Tarapaca-Chile
1Prof. Honorario de la UNMSM-Peru
Conferencistas locales
Renato Mario Benazic Tome UNMSMEugenio Cabanillas Lapa UNMSMVictor Rafael Cabanillas Zannini UNMSMLuis Enrique Carrillo Dıaz UNMSMPedro Celino Espinoza Haro UNIRoxana Lopez Cruz UNMSMJose Raul Luyo Sanchez UNMSMAlfonso Perez Salvatierra UNMSMMario Piscoya Hermoza2 UNMSMEnrique Vasquez Huaman3 Universidad del PacıficoEdgar Diogenes Vera Saravia UNMSM
2Profesor Emerito de la Universidad Nacional Mayor de San MarcosEx-Vice Rector Academico UNMSMEx-Decano FCM-UNMSM
3Director de Desarrollo de la Universidad del Pacıfico, profesor del DepartamentoAcademico de Economıa y miembro del Centro de Investigacion de esta casa de estudios.
VI SIMPOSIO INTERNACIONAL DE MATEMÁTICA APLICADA
20 al 24 de Julio de 2015
PROGRAMACIÓN
HORA LUNES MARTES MIÉRCOLES JUEVES VIERNES
10:00-10:35 E. Vera A. Pérez 09:35-10:35 J. Soriano
A. Rodriguez
10:35-11:35 M. da Silva A. Mércio C. Raposo R. Fuentes
11.35-12:35 X. Carvajal Ma To Fu P. Gamboa L. Lacerda
12:35-13:10 R. Benazic 12:35-13:35 O. Vera
V. Tapia J. Luyo
CLAUSURA
15:00-15:35 INAUGURACIÓN R. Cabanillas E. Cabanillas P. Espinoza
15:35-16:35 Conferencia Inaugural J. Muñoz
E. Hingant J. Pozo A. Rambaud
16:35-17:35 Intermédio Musical Artístico
Minicurso
Minicurso
Minicurso
17:35-18:35 Minicurso L. Carrillo
17:35 -18:10 R. López
H. Fernández 17:35-18:10 E. Vásquez
18:35-19:35 L. Fatori 18:10-19:10 V. Poblete
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 LUNES
15:00-15:35 INAUGURACIÓN
15:35-16:35 Conferencia Inaugural J. Muñoz
16:35-17:35 Intermédio Musical Artístico
17:35-18:35 Minicurso L. Carrillo
18:35-19:35 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 MARTES MIÉRCOLES JUEVES
10:00-10:35 E. Vera A. Pérez 09:35-10:35 J. Soriano
10:35-11:35 M. da Silva A. Mércio C. Raposo
11.35-12:35 X. Carvajal Ma To Fu P. Gamboa
12:35-13:10 R. Benazic 12:35-13:35 O. Vera
V. Tapia
15:00-15:35 R. Cabanillas E. Cabanillas P. Espinoza
15:35-16:35 E. Hingant J. Pozo A. Rambaud
16:35-17:35 Minicurso
Minicurso
Minicurso
17:35-18:35 17:35 -18:10 R. López
H. Fernández 17:35-18:10 E. Vásquez
18:35-19:35 18:10-19:10 V. Poblete
M. Cataldo
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 VIERNES
10:00-10:35 A. Rodriguez 10:35-11:35 R. Fuentes
11.35-12:35 L. Lacerda
12:35-13:10 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
Resumenes de las Conferencias
Indice
1. Semigrupos de Operadores Aplicados a Finanzas (Minicurso)Luis Enrique Carrillo Dıaz 1
2. Estabilidad de sistemas dinamicos con multiples mecanismos disi-pativosJaime Munoz Rivera 3
3. Exact controllability for Bresse system with variable coefficientsJuan Amadeo Soriano Palomino y Rodrigo Andre Schultz 5
4. Exact controllability for Bresse system with variable coefficientsX. Carvajal, P. Gamboa, O. Vera 8
5. Asymptotic Behaviour of Solutions to a system of coupled Schrodin-ger equationsX. Carvajal 11
6. The Lack of Exponential Stability to Boundary Dissipative PlatesL.H. Fatori 12
7. Existence and uniqueness of solution for a unilateral problem for theKlein-Gordon operator with Kirchhof-Carrier nonlinearityC. Raposo, D. Perira, G. Araujo, A. Baena 14
8. Energy decay of semilinear wave equations with moving boundaryMa To Fu 15
9. A natural slow-fast system arising in the scaling of the Becker-DoringequationsErwan Hingant 16
10.Taxas de decaimento para sistemas de Timoshenko nao-homogeneosfracamente dissipativos 17
11. Stability in Transmission Problems to Multicomponent TimoshenkoBeamsWith Localized Kelvin-Voigt DissipationJ. Munoz Rivera, A. Rambaud, O. Vera 18
12.Estudio del decaimiento exponencial para un problema de transmi-sion en termoelasticidad unidimensionalA. Perez Salvatierra 19
13.Riemann Solutions For CounterflowCombustion in Light Porous FoamL. Lacerda Fernandez, G. Chapiro 20
14.Rates of Decay for Hyperbolic ThermoelasticityHugo Fernandez Sare 21
15.No-Flux Boundary Problem Involving p(x)-Laplacian-Like Operatorsvia Topological MethodsEugenio Cabanillas Lapa 22
16. Singularidades Aisladas de Foliaciones por CurvasRenato Benazic Tome 23
17.Numerical Irrelevant Solutions (NIS) in Nonlinear Elliptic Eingen-value ProblemsPedro Espinoza Haro 24
18. Solucao das Equacoes de Saint Venant pelo Metodo das Caracterısti-cas usando SplinesA. Mercio Lobeiro, M. Vieira Passos, J. Soriano Palomino 26
19. Fractional Resolvent Families of Bounded SemivariationH. Henriquez, V. Poblete, J. Pozo 29
20.Estabilizacion de Sistemas de Control No Lineal Mediante el Princi-pio de ReduccionVictor Tapia Funes 32
21. ¿Porque Algebra Geometrica?Edgar Vera S. 33
22.Bifurcaciones en un modelo depredador-presa tipo Leslie-Gower conretardoRoxana Lopez Cruz 34
23.Global solutions and decay of a non linear coupled systemwith thermo-elasticRicardo Fuentes Apolaya 35
Semigrupos de Operadores Aplicados a Finanzas[Minicurso]
Luis Enrique Carrillo Dıaz*
Resumen
Actualmente los especialistas en finanzas, demandan nuevos modelos ma-tematicos, que les permitan tomar decisiones con el menor riesgo posible. Desdeel ano 1973, uno de los modelos mas requeridos es el de Black-Scholes [2], quie-nes ganaron el Premio Nobel de economıa en 1997. Estos modelos son empleadosen los Mercados de Opciones Financieras, lo cual esta generando el desarrollo deotras areas de la propia matematica, en la intencion de formular y hallar solucio-nes de nuevos modelos. A grosso modo, las opciones son contratos de compra oventa a futuro, lo cual lleva implıcita una cuota de incertidumbre, que se deseaminimizar sin detrimento de las ganancias. Como la dinamica de tales fenomenoses gobernada por movimientos Brownianos, aparecen comportamientos que nopueden ser representados con los metodos clasicos.
Este Minicurso, trata sobre un nuevo enfoque para Valoracion de Opciones, elcual hace uso de la Teorıa de Semigrupos de Operadores y su herramienta principal,el famoso teorema de Hille-Yosida. Se expone principalmente el protocolo paravaloracion de Opciones europeas y americanas. Respecto a las opciones asiaticasse establecen alternativas de solucion en situaciones especiales.
Aprovechando la analogıa entreOpciones Financieras yOpciones Reales, se abor-da brevemente el problema de Opciones Reales americanas mediante semigrupode operadores.
Contenido
1. Conceptos fundamentales.Breve introduccion a Opciones: Opciones Financieras y Opciones Reales.
2. Teorıa de Semigrupos: Teorema de Hille-Yosida.
3. Valoracion de Opciones Financieras vıa semigrupos de operadores.• Valoracion de Opciones europeas.• Valoracion de Opciones americanas.• Valoracion de Opciones asiaticas.
4. Valoracion de Opciones Reales con semigrupos de operadores.
5. Aplicaciones.
*Profesor Principal de la Facultad de Ciencias Matematicasde la Universidad Nacional Mayor de San Marcos
1
Referencias
[1] Belleni-Morante, A. andMcBride, A.C. Applied Nonlinear Semigroups; JohnWileyand Sons Ltda. (1998).
[2] Black, F. and M. Scholes, The Pricing of Options and Corporate Liabilities, Journalof Political Economy, (1973)
[3] Brandao-Cortazar, Evaluating Environmental Investments: A Real Options Ap-proach PUC-DEI, Rio de Janeiro, Brasil, (2000).
[4] Colombo, F.,Giuli, M. and Vespri, V., A semigroup approach to no-arbitrage pricingtheory: constant elasticity variance model and term structure models, Progress inNonlinear Differential Equations and Their Appl. Vol 55, 113-126, Birkhauser(2003).
[5] Cruz-Baez, D.I., & Gonzalez-Rodrıguez, J.M., Semigroup Theory Applied to Op-tions. Hindawi Publishing Corporation Journal of Applied Mathematics, 2-3(2002) 131–139. Copyright (2002)
[6] Guimaraes Dıas, M. A., Analise de Investimentos com Opcoes Reais. Teorıa e Praticacom Aplicacoes em Petroleo 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 InThe 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 Ma-nagement Science, pp. 141–183. (1973).
[10] Munoz Rivera, J., Estabilizacao de Semigrupos e Aplicacoes. Laboratorio Nacionalde Computacao Cientifica. Petropolis, Rio de Janeiro-Brasil (2009).
[11] Samanez, C.P.; Ferreira, L. ; Do Nascimento, C. and Bisso, C., Eva-luating the economy embedded in the Brazilian ethanol–gasoline flex-fuel car: a Real Options approach,, Vol. 46, No. 14, 1565–1581,http://dx.doi.org/10.1080/00036846.2013.877573 8 Applieds Econo-mics(2014).
[12] Sick, G. and Gamba, A., Some Important Issues Involving Real Options: An Over-view, University of Calgary, (2005).
[13] Trigeorgis, L. Real Options: Managerial Flexibility and Strategy in Resource Alloca-tion, MIT, Press, (1996)
2
Estabilidad de sistemas dinamicos conmultiples mecanismos disipativos
Jaime E.Munoz RiveraLaboratorio Nacional de Computacion CientıficaRua Getulio Vargas 333, CEP 25651-075, RJ Brasil
Instituto de Matematica-UFRJ
Resumen
En este trabajo estudiaremos los efectos de diversos mecanismos disi-pativos. Nuestra intension es mostrar, que la suma de estos mecanismosno mejora en general la estabilizacion y puede suceder todo lo contrario,esto es, que la estabilizacion sea muy lenta. En esta conferencia mostrare-mos como el orden de los mecanismos puede ocasionar diferentes tasas dedecaimiento. Adicionalmente mostraremos como se deben ordenar estosmecanismos de tal forma de optimizar la tasa de decaimiento. Los meca-nismos que consideraremos en esta exposicion son: Mecanismo viscoso dela clase de Kelvin-Voight, mecanismos tipo friccional, mecanismos termi-cos. En los siguientes graficos mostramos el caso de tres componentes, condos mecanismos disipativos.
3
Este trabajo fue realizado con la colaboracion de Mauricio Sepulveda, Univer-sidad de Concepcion - Chile, Octavio Vera Villagran, Universidad del Bio-Bio- Chile y Margareth Alves, Universidad de Visoca - 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 withlocally distributed Kelvin-Voigt damping. SIAMJournal of Control and Optimiza-tion Vol. 36. 31086-1098 (1998).
[3] M. Alves, J.E. Munoz Rivera, Mauricio Sepulveda and O. Vera Villagran: Thelack of exponential stability in certain transmission problems with localize kelvin-voigt dissipation. SIAM Journal of Applied Mathematics Volume 74, Numero 2,paginas 345 - 365 (2014).
[4] F. Ammar Khodja, A. Benabdallah, J.E. Munoz Rivera and R. Racke: Energydecay for Timoshenko systems of memory type, J. Differential Equations, 194-82–115-(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 linearevolution 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] Munoz 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 operatorsemigroups, Math. Nachr. Vol. 279, (1), pages 1425-1440, (2006).
[11] Soufyane, A.: Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci. Paris, Ser.I 328, 731–734, (1999).
[12] Timoshenko, S. P.: On the correction for shear of the differential equation for trans-verse vibrations of prismatic bars. Philosophical Magazine, 6, 744-746;3, (1921).
4
Exact controllability for Bresse system withvariable coefficients
Juan Amadeo Soriano Palomino Rodrigo Andre SchulzDepartamento de Matematica-DMA, UEM,
87020-900, Maringa, Avda Colombo, 5790, Campus Universitario
Maringa, PR, (44)30114040
E-mail: [email protected]
Abstract
This paper is concernedwith the internal exact controllability of a gen-eralized Bresse system with variable coefficients, which the controls func-tions acts in an arbitrarily small subinterval (l1, l2) of (0,L). Our computa-tion suggests a minimal time control and a region where the controls aremore effective. The variable coefficients can be viewed as a generalizationof Laplacian operator. The main result is obtained by applying HilbertUniqueness Method proposed by Lions, without using the Holmgren’suniqueness theorem or the hypothesis of equal-speed waves of propaga-tion.
Introduction
Consider the Bresse system given by
ρ1ϕtt − k(a(x)ϕx +ψ + lω)x − k0l[ωx − lϕ] = h1χ,ρ2ψtt − (b(x)ψx)x + k(ϕx +ψ + lω) = h2χ,ρ1ωtt − k0[c(x)ωx − lϕ]x + kl(ϕx +ψ + lω) = h3χ,
(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), (2)
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.
(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) andh3 = 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 con-trollability of (1)− (3).
1 Assumptions
In the observability and controllability results, we always take T > 2αR where
α =max1,ρ1k,ρ2b0,ρ1k0
, (4)
and
R =maxl1,L − l2
, (5)
2 Main result
Theorem 1Assume that a,b,c ∈ W 1,∞(0,L) satisfying a(x) ≥ 1,b(x) ≥ b0 > 0, and c(x) ≥1in(0,L). Let T > 2αR,α, and R given in (4) and (5), ϕ0,ψ0,ω0 ∈ H1
0 (0,L) andϕ1,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
ρ1ϕtt − k(a(x)ϕx +ψ + lω)x − k0l[ωx − lϕ] = h1χ, in (0,L)× (0T ),ρ2ψtt − (b(x)ψx)x + k(ϕx +ψ + lω) = h2χ, in (0,L)× (0T )ρ1ωtt − k0[c(x)ωx − lϕ]x + kl(ϕx +ψ + lω) = h3χ, in (0,L)× (0T )ϕ(0) = ϕ(L) = ψ(0) = ψ(L) = ω(0) = ω(L) = 0, in (0,L)× (0T )ϕ(x,0) = ϕ0(x),ϕt(x,0) = ϕ1(x), x ∈ (0,L),ψ(x,0) = ψ0(x),ψt(x,0) = ψ1(x), x ∈ (0,L),ω(x,0) =ω0(x),ωt(x,0) = ω1(x), x ∈ (0,L),
(6)
where χ is the characteristic function of interval (l1, l2) checks
ϕ(x,T ) = 0, ϕ(x,T ) = 0,ψ(x,T ) = 0, ψ(x,T ) = 0,ω(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, Munoz Rivera J.E, Fatori L.H.: Bresse system with indefinitedamping. Journals of Mathematical Analysis and Applications; 387: 284-290, (2011).
[3] Alabau Boussouira F, Munoz Rivera J.E, Almeida Junior D.S.: Stabilityto weak dissipative Bresse system. Journals of Mathematical Analysis andApplications; 374(2), 481-498, (2011).
[4] Charles W, Soriano J.A, Falcao Nascimento F.A, Rodrigues J.H.: Decayrates for Bresse system with arbitrary nonlinear localized damping. Journalof Differential Equations; 8:2267-2290, (2013).
[5] Charles W., Soriano J.A, Schulz R.A.: Asymptotic stability for Bresse sys-tem. Journal of Mathematical Analysis and Applications; 412(1), 369-380,(2014).
7
Exact controllability for Bresse system withvariable coefficients
X. Carvajal, P. Gamboa∗, & O. Vera†
Abstract
In this work we find exact solutions to the fifth-order KDV-BBM typemodel that appear to describe the propagation of long waves in shallowwater. We study the possibility of blow-up phenomenon of the fifth-orderKDV-BBM type model under certain restrictions on the coeffcients. More-over, by applying the Ince transformation we also establish exact travel-ling 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 fifthorder BBM-KdV type equation
ηt + ηx − 1
6ηxxt + δ1ηxxxxt + δ2ηxxxxx +34
(η2
)x+γ
(η2
)xxx− 1
12
(η2x
)x− 1
4
(η3
)x= 0,
η(x,0) = η0(x)(1)
where η = η(x; t) is a real-valued function, and δ1 > 0, δ2;γ ∈ R. This modelwas recently introduced by Bona et al [1] to describe the unidirectional propa-gation of water waves. It was formally obtained as a second order approxima-tion from the higher order generalized Boussinesq system derived by Bona etal [2], which describes the two-way propagation of water waves.
Finally we consider an equation of Benney-Lin type, that is,
ut +λ1uxxxxx +λ2uxxxx +uxxx +λ3uxx +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 knownas Benney-Lin equation.
ut +λ1uxxxxx +λ2(uxxxx +uxx) +uxxx +uux = 0;x ∈R; t > 0; (3)
∗Instituto de Matematica, Universidad Federal de Rio de Janeiro, Av. Athos da SilveiraRamos, P.O. Box 68530, CEP:21945-970, RJ. Brazil E-mail address: [email protected] E-mailaddress: [email protected]†Departamento de Matematica, Universidad del Bıo Bıo, Collao 1202, Casilla 5-
C,Concepcion. Chile E-mail address: [email protected]
8
where u = u(x; t) is an unknown real-valued function, λ1,λ2 ∈ R and λ2 > 0. Itdescribes the propagation of one-dimensional small but finite amplitude longwaves 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
)δ125650δ2 = 190;
then an exact solution of (1) is
η(x; t) =sech2(kx −ωt)
(1− tanh(kx −ωt)−
√3
2|k| sech(kx −ωt))2 +α; (4)
where C+k ;C−k are constants that take values either 0 or 1, then,
limx→±∞η(x; t) = α − 4k
2
3C±k . (5)
2. Blow-up phenomena
We start by recalling the concept of the blow-up solution. Let T be the maxi-mal time of existence of the solution η(x; t). We say that the solution η has theblow-up property in the space X if and only if
supt∈[0,T )
‖η(t)‖X =∞.
We say that the solution η does not have blow-up property in the space Xif
supt∈[0,T )
‖η(t)‖X <∞.
The solution in (4) have singularity along the line
s(t) =ωkt − lnk0
k, t ≥ 0, (6)
whereω
k≈ 1.54978 − lnk0
k≈ 0.2492.
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)
to the IVP (1). If δ1 > 0 and γ ≤ 142
, then the corresponding solution blows-up
in H4 if and only if
liminft→T −
infx∈R
ηx(x,t) = −∞ or limsupt→T −
supx∈R
ηx(x,t) =∞ (7)
As our solution satisfies supx∈R |η(x; t)| = ∞, we concludes that η(x; t) haveblow-up in H4.
9
References
[1] Bona, J. L., Carvajal, X., Panthee, M., Scialom, M.: Higher-order models forunidirectional water waves, preprint, 1-31.
[2] Bona J. L., Chen M. and Saut J.-C.: Boussinesq equations and other systemsfor small-amplitude long waves in nonlinear dispersive media I. Derivationand linear theory, J. Nonlinear Sci. 12, 283-318, (2002).
10
Asymptotic Behaviour of Solutions to asystem of coupled Schrodinger equations
Xavier Carvajal∗
Abstract
This paper is concerned with the behaviour of solutions to a system ofcoupled Schrodinger 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 inmany physical problems, especially in nonlinear optics. When the so-lution there exists globally we obtain the growth of the solutions in theenergy space. Also we find some conditions in order to obtain blow-up inthis space.
This work is jointly with Pedro Gamboa.
∗Instituto de Matematica, Universidad Federal de Rio de Janeiro, Av. Athos da SilveiraRamos, P.O. Box 68530, CEP:21945-970, RJ. Brazil E-mail address: [email protected]
11
The Lack of Exponential Stability toBoundary Dissipative Plates
L. H. Fatori ∗
Departamento de Matematica, Universidade Estadual de Londrina
86051-990 Londrina, PR, Brazil
J. E. Munoz Rivera †
Laboratorio de Nacional de Computacao Cientıfica, LNCC/MCT
25651-070 Petropolis, RJ, Brazil
and
Instituto de Matematica, 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, withdissipativemechanism effective in the interior of the domain and/or dissi-pative boundary condition. More specifically, letΩ be a bounded domainof Rn types star-shape with smooth boundary ∂Ω = Γ0 ∪ Γ1 where Γ0 andΓ1 are closed sets, disjoint and not empty of ∂Ω, as shown below.
and we consider the initial-boundary value problem
utt −γ∆utt +α∆2u + a(x)ut = 0, in Ω ×R+, (1)
with boundary conditions
u =∂u
∂ν= 0, on Γ0 ×R+ (2)
∆u = 0, −γ ∂utt∂ν
+α∂∆u∂ν
= kut , on Γ1 ×R+ (3)
∗Email: [email protected].†Email: [email protected].
12
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. Ourmain 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 spec-trum radius of an operator S is invariant by compact perturbations. Moreover,from Borichev and Tomilov Theorem, we prove that the solution decays poly-nomially (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 nonlinearbeam under boundary output feedback control, Nonlinear Anal. 66 427-441,(2007).
[5] J. E. Lagnese:, Boundary Stabilization of Thin Plates, SIAM Studies in Ap-plied Mathematics, vol. 10, SIAM, Pennsylvania, (1989).
[6] J. Lagnese, G. Leugering, Uniform stabilization of a nonlinear beam by non-linear boundary feedback, J. Differential Equations 91, (1991), 355-388.
[7] J.L. Lions,: Quelques Methodes de Resolution des Problemes aux Limites NonLineaires, Dunod Gauthier-Villars, Paris, (1969).
[8] M. Reed, B. Simon,: Methods of Modern Mathematical Physics IV: Anal-ysis of Operators, Academic Press, San Diego, (1978).
[9] S. Woinowsky-Krieger,: The effect of an axial force on the vibration of hingedbars, J. Appl. Mech. 17 35-36, (1950).
[10] Borichev, A., Tomilov, Y.,: Optimal polynomial decay of functions and oper-ator semigroups, Mathematische Annalen 347(2), 455–478, (2009).
13
Existence and uniqueness of solution for aunilateral 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 Klein-Gordon
L =∂2u
∂t2−M(|∇u |2)∆u +M1(|u |2)u − f .
Using an appropriate penalization, see [1] and references therein, we ob-tain a variational inequality for the equation of Klein-Gordon perturbedand then the existence and uniqueness of solutions is analyzed.
Acknowledgement. Wewould like to express our gratitude to the FAPEMIG- Fundacao de Amparo a Pesquisa do Estado de Minas Gerais.
References
[1] C. A. Raposo, D. C. Carvalho, G. M. Araujo and A. Baena. Unilateral Prob-lems For the Klein-Gordon operator with nomliarity of Kirchhoff-Karriertype. Electronic Journal of Differential Equations, Vol. 2015, pp. 1–14,(2015).
∗Department of Mathematics, Federal University of Sao Joao del-Rei. Sao Joao del-Rei -MG 36307-352, Brazil [email protected]†Department of Mathematics, State University of Para. Belem - PA 66113-200, Brazil duci-
[email protected]‡Department of Mathematics, Federal University of Para Belem - PA 66075-110, Brazil
[email protected]§Department of Mathematics, Federal University of Para Belem - PA 66075-110, Brazil
14
Energy decay of semilinear wave equationswith moving boundary
To Fu MaInstituto de Ciencias Matematicas e de Computacao
Universidade de Sao Paulo13566-590 Sao Carlos, SP, Brazil.
Abstract
This talk is dedicated to the energy stability of weakly damped semi-linear wave equations defined on domains with moving boundary. Sincethe boundary is a function of the time variable, the problem is intrinsi-cally non-autonomous. Under the hypothesis that the lateral boundaryis time-like, the solution operator of the problem generates an evolutionprocess U(t,τ) : Xτ → Xt , where Xt are time-dependent Sobolev spaces.Then, for non-contracting domains, we discuss the exponential stabilityof 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 dependentdomain, J. Math. Anal. Appl. 42 (1973) 29-60.
[3] P. E. Kloeden, J. Real and C. Sun, Pullback attractors for a semilinear heatequation on time-varying domains, J. Differential Equations 246 (2009)4702-4730.
15
A natural slow-fast system arising in thescaling of the Becker-Doring equations
Erwan HingantCi2ma - Universidad de Concepcion, Chile
joint work with Julien Deschamps and Romain Yvinec
Abstract
We will present the mathematical connection between two classicalmodels of phase transition phenomena describing different stages of clus-ter growth, namely, the Becker-Doring equations (BD) and the Lifshitz-Slyozov equation (LS). The former consist in an infinite set of ODE, onefor each size of clusters. While, the latter is a PDE on the density functionaccording to a continuous-size variable. Suitable scaling of BD with re-spect to a small parameter ε has been studied in [1, 4] where the authorsrigorously derive LS when the parameter ε→ 0. In [2] we derive the samelimit in a stochastic context remarking that an underlying system on thesmall size of clusters behave at a different time scale than the equation onthe density function. In the spirit of the works started by Fenichel in the70’s, see e.g. [3], we take advantage of this sub-system to derive variousboundary conditions on the LS equation which were lacking in previousworks. Here, we focus on the deterministic version of this result.
References
[1] Jean-Francois Collet, Thierry Goudon, Frederic Poupaud, and AlexisVasseur. The Beker-Doring system and its Lifshitz-Slyozov limit. SIAMJournal on Applied Mathematics, 62(5):1488–1500 (electronic), (2002).
[2] Julien Deschamps, Erwan Hingant, and Romain Yvinec. From a stochas-tic Becker-Dring model to the Lifschitz-Slyozov equation with boundaryvalue. Preprint arXiv:1412.5025, (2014).
[3] Christian Kuehn. Multiple Time Scale Dynamics, volume 191 of AppliedMathematical Sciences. Springer International Publishing, Cham, (2015).
[4] Philippe Laurencot and Stephane Mischler. From the Becker-Doringto the Lifshitz-Slyozov-Wagner equations. Journal of Statistical Physics,106(5-6):957–991, (2002).
16
Taxas de decaimento para sistemas deTimoshenko não-homogêneos fracamente
dissipativos
Marcio A. Jorge da Silva∗
ResumoNesta conferência serão abordados resultados sobre a existência e
taxas de decaimento para sistemas vigas de Timoshenko não homogê-neos fracamente dissipativos. Neste caso, os coeficientes são funçõesnão constantes que podem variar de acordo commaterial que compõetais sistemas. Sendo assim, para a estabilização exponencial dos sis-temas estudados, uma igualdade local das velocidades de propagaçãode onda são consideradas como hipóteses. Quando tal condição localnão necessariamente vale, então decaimento polinomial é mostradopara os sistemas de Timoshenko em geral.
Referências
[1] F. Ammar-Khodja, S. Kerbal and A. Soufyane, Stabilization of the no-nuniform Timoshenko beam, J. Math. Anal. Appl. 327, 525-538,(2007).
[2] J. E. Muñoz Rivera and A. I. Ávila, Rates of decay to non homogeneousTimoshenko model with tip body, J. Differential Equations 258 , no. 10,3468-3490, (2015).
[3] A. Soufyane, Exponential stability of the linearized nonuniform Ti-moshenko 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 toMulticomponent Timoshenko Beams With
Localized Kelvin-Voigt Dissipation
J. Munoz-Rivera∗& A. Rambaud†& O. Vera‡
[email protected]; [email protected] ; [email protected]
Abstract
We consider the transmission problem of Timoshenko’s beam com-posed of N components, each of them being either purely elastic (E), or aKelvin-Voigt viscoelastic material (V), or another elastic material insertedwith a frictional damping mechanism (F). Such material is illustrated inFigure 1 for N = 7. We prove that the transmissionproblem is always well-posed. Our main result is that the rate of decay depends on the position ofeach component. More precisely, we prove that the beam is exponentiallystable if and only if all the elastic components have at least one frictionalneighbouring material. Otherwise, the decay is only polynomial of order1/t2.
———————-Work partially supported by Fondecyt Project 11130378 and GIMNAP,
Depto de Matematica, Universidad del Bio-Bio.∗LNCC, Rio de Janeiro, Brasil†Universidad del Bio-Bio, Concepcion‡Universidad del Bio-Bio, Concepcion
18
Estudio del decaimiento exponencial paraun problema de transmision entermoelasticidad unidimensional
Alfonso Perez Salvatierra
UNMSM [email protected]
Abstract
En el presente trabajo estudiamos la existencia, unicidad de solucion yel decaimiento exponencial de la energıa asociada al sistema de un prob-lema de transmision en termoelasticidad unidimensional representadapor,
utt −αuxx +mθx + f (u) = h1, en (L1,L2)× (0,∞)θt − kθxx +muxt = h2, en (L1,L2)× (0,∞)vtt − bvxx = h3, en (0,L1)× (0,∞)
(1)
Condiciones de frontera:
u(0, t) = θ(0, t) = v(L2, t) = 0u(L1, t) = v(L1, t), αux(L1, t)−mθ(L1, t) = bvx(L1, t)θx(L1, t) = 0
(2)
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) = v0(x), vt(x,0) = v1(x), x ∈ (0,L1)
(3)
El cuerpo esta compuesto por dos partes, una parte elastica y la otra partetermoelastica. La prueba de la existencia y unicidad de la solucion al sistemase garantiza por el teorema de Lummer-Phillips y la estabilidad exponencialpor criterios del resolvente del operador matriz A(ρ(A).
References
[1] M. Alves, J.E. Munoz Rivera, Mauricio Sepulveda and O. Vera Villagran:The lack of exponential stability in certain transmission problems with localizekelvinvoigt dissipation. SIAM Journal of Applied Mathematics Volume 74,Numero 2, paginas 345 - 365 (2014).
19
Riemann Solutions For CounterflowCombustion in Light Porous Foam
Laura Senos Lacerda Fernandez∗ & Gigori Chapiro†
Abstract
In this talk we will consider a system of three evolutionary partialdiferential equations that models combustion of light porous foam underair injection. We will see how to reduce this problem to an EDO by achange of variables and we will study the existence of the combustionwave sequences with negative velocity appearing in Riemann solutions.
References
[1] G. Chapiro, L. Senos; Riemann solutions for counter flow combustion in lightporous foam, preprint.
[2] G. Chapiro, D. Marchesin, and S. Schecter; Combustion waves and Rie-mann 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, MG36036-900, 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 HyperbolicThermoelasticity
Hugo D.Fernandez Sare∗
Abstract
We study models in thermoelasticity involving non-classical theoryfor heat conduction. Results about stability of solutions for these systemswill be formulated.
∗Instituto de Matematica. Universidade Federal do Rio de Janeiro - Brasil
21
No-Flux Boundary Problem Involvingp(x)-Laplacian-Like Operators via
Topological Methods
Eugenio Cabanillas Lapa∗
Abstract
The purpose of this article is to obtain weak solutions for a class non-linear elliptic problem for the p(x)-Laplacian-like operators under no-fluxboundary conditions. Our result is obtained using a Fredholm-type resultfor a couple of nonlinear operators and the theory of the variable expo-nent Sobolev spaces.
∗Universidad Nacional Mayor de San Marcos-Lima-Peru
22
Singularidades Aisladas de Foliaciones porCurvas
Renato Benazic Tome*
Resumen
En esta conferencia hablaremos sobre puntos singulares aislados de unsistema de n ecuaciones diferenciales complejas. Estudiaremos los prin-cipales invariantes y las formas normales cuando n = 2 y enumeramosalgunos resultados que pueden ser extendidos a dimensiones mayores a2.
*Universidad Nacional Mayor de San Marcos-Lima-Peru
23
Numerical Irrelevant Solutions (NIS) inNonlinear Elliptic Eingenvalue Problems
Pedro C. Espinoza Haro∗
Abstract
Consider the following the following kinds of nonlinear elliptic eigen-value problems: −∆u(x) = λf (u(x)) x ∈Ω
u = 0, in ∂Ω(1)
where Ω is a bounded open subset in Rn and whose border ∂Ω is smoothand
a)f : [0,∞)→R, es localmente Lipschitz continuab)f has exactly 2m non-negatives zeros s0 = 0 < s1 < · · · < s2m−1and sig[f (t)] = (−1)i , ∀t ∈ (si , si+1), i = 0,1, . . . ,2m− 1
(2)
We will say that f in (2) satisfies the “positive area condition”
F (s2i+1)− F (s2i−1) > 0 (3)
for each i = 1,2, . . . ,m− 2, where F(t) =∫ t0 f (s)ds.
The discrete analogue of (1) by Finite Difference whit suitable gridpoints at is
Ax = λh2f (x), x ∈Rn (4)
where A is aM-matrix, [9], [10], obtained in the discretization of operator−∆, h is themesh size and f (x) = (f (x1), . . . , f (xn)) is the Nemitskii operatorassociated to scalar function f . In this note we study the numericallyirrelevant solutions (NIS) of the problem (1), which do not approximateanalytical 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 andtheory of bifurcations. In this note we obtain, for the one-dimensionalcase, some features of the NIS.
∗Universidad Nacional de Ingenierıa-FIIS-Seccion de Posgrado.Ex-docente de la FCM-Universidad Nacional Mayor de San Marcos-Lima-Peru
24
References
[1] E. Bohl: On the bifurcation diagram of discrete analogues for ordinary bifur-cation problems, Math. Methods Appl. Sci., v. 1, pp. 566-671, (1979).
[2] W.-J. Beyn & J. Lorenz: Spurious solutions for discrete superlinear boundaryvalue problems, Computing, v. 28, pp. 42-51 (1982).
[3] AK. J. Brown y H. Budin: On the existence of positive solutions for a classof 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 non-linear 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 nonlinearelliptic 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 problemsversus their finite aproximation: numerically irrelevant solutions, J. ReineAngew Mathematik 322, 74-117, (1981).
[7] H. O. Peitgen y K. Schmitt: Positive and spurious solutions of Nonlin-ear eigenvalue problems, Springer Lectures Notes in Math. 878 275-324,(1981).
[8] H. Jurgens, H.O. Peitgen and D. Saupe: Topological perturbations in the nu-merical study of nonlinear eigenvalue and bifurcation problems, ProceedingsSymposium on Analysis and Computation of Fixed Points, S. M. Robin-son (ed.), New York-London, (1979).
[9] J. Schroder: M-matrices and generalizations using and operator theory ap-proach, 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 peloMétodo das Características usando Splines
Adilandri Mércio Lobeiro∗ Marlon Vieira Passos†
Juan Amadeo Soriano Palomino‡
[email protected]@uem.br
Resumo
O presente trabalho apresenta a solução numérica das equações deSaint Venant conjugando o Método das Características com InterpolaçõesCúbicas com splines naturais no lugar de Interpolações Lineares usual-mente adotadas para encontrar a velocidade média e o perfil da onda eminstantes de tempo pre-fixados.
1 Introdução
Na Engenharia Hidráulica, as equações de Saint Venant são frequentementeusadas em estudos de escoamento não permanente em canais. No caso parti-cular de canais retangulares de grande largura, as equações são
∂h∂t +u ∂h
∂x +∂u∂x = 0 (1)
∂u∂t +u ∂u
∂x + g ∂h∂x = g(S0x − Sfx ), (2)
em que u(x,t) é a velocidade média do escoamento (m/s) na direção x; h(x,t) éa 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 declividadelongitudinal e Sfx é a declividade da resistencia hidráulica na direção x, dadopor Sfx = |u|u/C2h, onde C é a constante de Chézy.
∗Departamento de Matemática, DAMAT, UTFPR87301-899, 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 Mé-todo das Caracterásticas, que é um método consagrado por transformar umsistema de Equações Diferenciais Parciais (EDPs) em um sistema de EquaçõesDiferenciais Ordinárias (EDOs) [1]. Neste caso, as equações (1) e (2) foramtransformadas em
dx
dt= u + c, (3)
d
dt(u +2c) = g(S0x − Sfx ), (4)
dx
dt= u − c, (5)
ddt
(u − 2c) = g(S0x − Sfx ). (6)
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 400m decomprimento, 5m de altura, 1m de largura, declividade S0x = −0.0016 e
(Sfx
)LP
=
0.5(uL|uL|/C2hL +uP |uP |/C2hP
), onde C = 100m(1/2)/s. Inicialmente o canal es-
tava cheio de água e a mesma encontrava-se parada, ou seja, a velocidade ini-cial era zero. Considerou-se a descarga a esquerda, Figura 1, dada pela funçãovazão qP , definida por
qP : R+ −→ R
t 7−→ qP(t) =
−0.1t se t ≥ 0 e t < 60−6+0.1(t − 60) se t ≥ 60 e t < 80
−4 se t ≥ 80.(7)
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 dascaracterísticas, ou seja, encontrar a velocidade média e a altura da água emqualquer ponto do canal no decorrer do tempo.
Ao aplicar o Método das Características nestas equações, é de praxe utilizara interpolação linear para encontrar a velocidade e a profundidade da onda empontos não conhecidos da malha construída para obter a solução numérica [2].A interpolação linear consiste em unir um conjunto de pontos com uma sériede linhas retas. Uma desvantagem desta aproximação é que não há diferencia-ção nos extremos de cada intervalo [3]. Para sanar esta dificuldade utilizou-sea interpolação com Spline Cúbico Natural.
27
2 Resultados principais
Ao utilizar o Spline Cúbico Natural, permitiu-se obter a profundidade e ve-locidade do escoamento em posições específicas ao longo do comprimento docanal e em instantes de tempo pré-fixados, o que tornou possível estimar taisvalores em qualquer ponto do canal, por meio de uma função duas vezes con-tinuamente diferenciável. Sua utilização também otimizou o código teóricopor, entre outros fatores, não haver a necessidade de um número grande desubdivisões no intervalo de comprimento estudado, uma clara vantagem secomparada com a Interpolação Linear, que é comumente utilizada.
Referências
[1] García-Navarro, P.; Brufau, P.; Burguete, J.; Murillo, J.: The Shallow Wa-ter Equations: An Example of Hyperbolic System. Monografías de La RealAcademia de Ciencias de Zaragoza, 31, 89-119, (2008).
[2] Lobeiro, A. M.: Solução das Equações de Saint Venant em uma e duas dimen-sões usando o Método das Características. Universidade Federal do Paraná,(2012).
[3] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P.: NumericalRecipes. The Art of Scientific Computing. New York: Cambridge UniversityPress, (2007).
28
Fractional Resolvent Families of BoundedSemivariation
Hernan R. Henrıquez∗, Veronica Poblete†& Juan C. Pozo‡
Abstract
In this work we establish the existence of α-resolvent families of boundedsemivariation for all 0 < α < 2. We show that theory of cosine operator familiesof bounded semivariation is a singular case of the theory of α-resolvent families.Furthermore, by using the α-resolvent families of bounded semivariation and ap-propriate conditions on the forcing function, we study the existence of strongsolutions of non-homogeneous fractional differential equations. We consider theautonomous and the non-autonomous cases.
Let X be a Banach space and suppose that A(t) :D(A(t)) ⊆ X→ X are closed linearoperators with domain D(A(t)) = D for all t ∈ [0;a];a > 0. We consider the followingproblem
Dαt u(t) = A(t)u(t) + f (t,u(t)), t ∈ [0,a],
u(0) = x,u ′(0) = y.
(1)
where α ∈ (1;2), and the fractional derivative Dαt is understood in the Caputo sense.
If A(t) = A for all t ∈ [0;a], the problem 1 is known in the literature by fractionalabstract Cauchy problem associated to A of order α. The existence of solutions of thisproblem is strongly related with the concept of α-resolvent family Sα(t)t≥0; intro-duced by Pruss [5] and widely developed by Bazhlekova [2]. In fact, the fractionaldifferential equation 1 is well posed if and only if A is the infinitesimal generator ofan α-resolvent family Sα(t)t≥0. For more information see [[5], Proposition 1.1].
In the autonomous case, we assume thatA generates an α-resolvent family Sα(t)t≥0and define the operators Pα(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 Matematica, Santiago-Chile.Partially supported by FONDECYT 1130144 and DICYT-USACH.†Universidad de Chile, Facultad de Ciencias, Santiago-Chile.
Partially supported by PAIFAC 2015.‡Universidad Catolica de Temuco, Departamento de Matematicas y Fısica, 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 theother hand, it has been showed in [3] that for α = 2 the conditions (i) and (ii) of Lemma0.1 are in turn equivalent to A be a bounded linear operator. In what follows we willshow that for 1 < α < 2 there are α-resolvent families of bounded semivariation gen-erated by unbounded operators. The following result establishes the Banach spaceswhere α-resolvent family of bounded semi-variation can be defined. It is a general-ization 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 α-resolvent family Sα(t)t≥0 of bounded semivariation on [0,a] for all a > 0, then theoperator 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 Agenerates 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 denoteU(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, de-noted byU(t; s) andW (t; s), respectively, andU ;W : ∆→L(X) are strongly continuousoperator valued maps. Moreover,
U(t; s)x = Sα(t − s)x +∫ t
sPα(t −ψ)B(ψ)U(ψ; s)xdψ; x ∈D(A);
U(t; s)x = Sα(t − s)x+∫ t
sPα(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 solu-tion given by
u(t) =U(t;0)x +∫ t
0U(ψ;0)ydψ +
∫ t
0Q(t; s)f (s)ds;
where
Q(t; s)z =∫ t
sgα−1(τ − s)U(t;τ)zdτ; z ∈ X; 0 ≤ s ≤ t ≤ a.
30
References
[1] J. B. Baillon, Caractere borne de certains generateurs de semi-groupes lineaires dansles espaces de Banach, C. R. Acad. Sci. Paris 290 ,757-760,(1980).
[2] E. G. Bazhlekova, Fractional Evolution Equations in Banach Spaces, EindhovenUniversity of Technology, Eindhoven, . Dissertation,(2001).
[3] D. Chyan, S. Shaw, S. Piskarev, On maximal regularity and semivariation of cosineoperator 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 semi-variation, and the inhomogeneous Cauchy problem, J. Evol. Equ. 8 (2), 283-305,(2008).
31
Estabilizacion de Sistemas de Control NoLineal Mediante el Principio de Reduccion
Vıctor Tapia Funes*
Resumen
En el presente trabajo se expone las condiciones para la estabilizacionde sistemas de control no lineal, mediante el principio de reduccion, endonde la teoria de la variedad central no se puede aplicar.
*Universidad de Tarapaca - Sede Iquique
32
¿Porque Algebra Geometrica?
Edgar Vera Saravia*
Resumen
El Algebra Geometrica fue creada por Clifford entre los anos 1873 y1879 como una generalizacion de los Cuaterniones de Hamilton. En estacharla comentaremos el aspecto unificador de esta estructura y su empleoen la fundamentacion matematica de la fısica y sus aplicaciones en otrasareas.
*Departamento de MatematicaUniversidad Nacional Mayor de San Marcos
33
Bifurcaciones en un modelodepredador-presa tipo Leslie-Gower con
retardo
Roxana Lopez-Cruz
Universidad Nacional Mayor de San Marcos, [email protected]
ResumenEste trabajo trata acerca del modelo depredador-presa tipo Leslie-Gower
modificado (1), teniendo en cuenta que la poblacion de presas es afectadapor un efecto Allee debil, ası como por un retardo τ en el crecimiento dela poblacion presa.
Xµ :
dx
dt=
(r
(1− x(t − τ)
K
)(x −m)− qy
)x
dy
dt= s
(1− y
nx
)y
(1)
Usamos el retardo τ como un parametro de bifurcacion. Demostramos laaparicion de una bifurcacion de Hopf, cuando el retraso discreto cruzacierta magnitud crıtica.
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, WorldScientific, (1998)
[3] C. C¸ elik, Hopf bifurcation of a ratio-dependent predator-prey systemwith time delay, Chaos, Solitons and Fractals 42,1474-1484,(2009)
[4] E. Gonzalez-Olivares, J. Mena-Lorca, A. Rojas-Palma, J. D. Flores,Dynamical complexities in the Leslie-Gower predator-prey model as con-sequences of the Allee effect on prey, Applied Mathematical Modelling35, 366-381, (2011)
[5] Y. Kuang, Delay differential equations with applications in PopulationsDynamics, Academic Press, Inc.(1993)
[6] H. Smith,An Introduction to Delay Differential Equations with SciencesApplications, Springer (2011)
34
Global solutions and decay of a non linearcoupled system with thermo-elastic
Ricardo Fuentes Apolaya
Universidade Federal Fluminense-Rio de [email protected]
Abstract
In this present work, the author prove the existence of global solutionsand the decay of nonlinear wave equation with thermo-elastic couplinggive by the system of equation:
u ′′(x, t)−µ(t)∆u(x, t) +n∑
i=1
∂θ∂xi
(x, t) +F(u(x, t)) = 0, in Q =Ω × (0,∞)
θ′(x, t)−∆θ(x, t) +n∑
i=1
∂u ′
∂xi(x, t) = 0 in Q
where u is displacement, θ is absolute temperature, ∆ denotes theLaplace operator, µ is a positive real function of t, F : R → R is conti-nous 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 |ρv usually appears in relativistic quantum
mechanic (see Segal [6] o Schiff [5]). Lions [3] studied the wave equa-tion with the same non linearity, i.e., |v |ρv, in a smooth bounded opendomain Ω of Rn and proved existence and uniqueness of solution usingboth Faedo-Galerkin’s and Compactnesss’ methods. In [1] investigatedthe system coupling with F(v) = |v |ρv . They established global existenceand strong and weak solutions by Faedo-Galerkin’s method using a basisof the space H1
0 (Ω)∩H2(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 uniqueglobal weak solution.
References
[1] R. Fuentes, H. Clark and A. Feitosa, On a nonlinear coupled system withinternal damping, Electronic Journal of Differential Equations, Volume2000, 64,1-17, (2000).
[2] H. Brezis and T. Cazenave, Non Linear Evolution Equations, Lecture Notesat. Instituto de Matematica, UFRJ, Rio de Janeiro, RJ, Brasil, (1994).
35
[3] J. L. Lions, Quelques methodes de resolution des problemes aux limites nonlineares, Dunod-Gauthier Villars, Paris, First edition, (1969).
[4] M. Milla and L. A. Medeiros, Hidden regularity for Semilinear HyperbolicPartial 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, 1-9,(1951).
[6] I. Segal, The global Cauchy problem for a relativistic scalar field with powerinteraction, Bull. Soc. Math., France, 91, 129-135, (1963).
[7] W. Strauss, On weak solutions of semilinear hyperbolic equations, An. Acad.Bras. Ciencias Volume 42, 04, 645-651, (1950).
36
Mapa de la Ciudad Universitaria de San Marcos
