Existencia global y explosión de la solución de un problema de difusión - reacción

Julio Becerra Saucedo

Resumen


En este artículo se hace un estudio analítico sobre la existencia global y local de la solución de un problema de difusión - reacción. Se demuestra que si la solución existe localmente entonces
esta llega a explotar en tiempo finito. Este resultado se extiende al caso en que la solución exista globalmente. Se llega a concluir que el tiempo máximo de existencia de la solución depende del dominio, del término que representa la reacción en la ecuación y de una función prueba definida en este trabajo. Así mismo se plantea la posibilidad de extender la existencia local a global usando el concepto de solución propia.


Palabras clave


problema de difusión - reacción; existencia global; existencia local; solución explosiva; tiempo de exposión

Texto completo:

PDF

Referencias


J.M. Arrieta, On boundedness of solutions of reaction - diffusion equations with nonlinear boundary conditions.

Proceedings of the American Mathematical Society. Feb. 2008; 136(1): 151 - 160.

J.M. Arrieta AND A. Rodriguez Bernal, Blow - up versus global boundedness of solutions of reaction -

diffusion equations with nonlinear boundaryconditions. Proceedings of Equations. 2005; 11: 1 - 7.

J.M. Arrieta AND Rodriguez Bernal A, Localization on the boundary of blow - up for reaction - diffusion

equations with nonlinear boundary conditions. Communications in Partial Differential Equations. 2004;

(7 - 8): 1127 - 1148.

k. Balazs, Semilinear Parabolic Problems [Tesis de Maestría]. Eotovos Lor´and University. Facultad de Ciencias;

J.M. Ball, Remarks on blow - up and non existence theorems for nonlinear evolution equations. Quartt. J.

Math Oxford. Mar. 1977 ; 2(28): 473 - 486.

P. Baras AND L. Cohen, Complete Blow - Up after Tmax for the Solution of a Semilinear Heat Equation.

Journal of Functional Analysis. 1987; 71: 142 - 174.

R. Bhadauria, A.K. Singh AND D.P. Singh, A Mathematical model to Solve Reaction Diffusion Equation

using Differential Transformation Method. International Journal of Mathematical Trends and Technology.

; 2(2): 26 - 31.

S. Chen AND D. Yu, Global existence and blow up solutions for quasilinear parabolic equations. Journal of

Mathematical Analysis and Applications 2007; 335: 151 - 167.

S. Chen AND W.R. Derrick, Global existence and blow up solutions for a semilinear parabolic system Rocky

Mountain Journal of Mathematics 1999. 29(2): 449-457.

T. Cazenave AND A. Haraux, An Introduction to Semilinear Evolution Equations. Clarendon Press - Oxford.

T. Cazenave, F. Dickstein AND F.B. Weissler, Universal solutions of a nonlinear heat equation on Rn.

Scuola Normale Superiore Di Pisa. Classe di Scienze (5). 2003; 2: 77 -117.

T. Cazenave, F. Dickstein AND F.B. Weissler, Global existence and blow up for sign changing solutions

of the nonlinear heat equation. Journal of Differential Equations. 2009; 246: 2669 - 2680.

M.J. Chadam, A. Pierce AND Y. Hong - Ming,. The blow - up property of solutions to some diffusion

equations with localized nonlinear reactions. Journal Mathematic Analitic and Applications. 1992; 160:

- 328.

G. De Assis, Soluciones globales uniformementes limitadas para una ecuación de calor semilineal [Tesis de Maestría]. Universidad de Brasilia - Instituto de Ciencias Exactas; 2012.

A. De Pablo, An introduction to the problem of blow - up for semilinear and quasilinear parabolic equations.

MAT - Serie A. 2006; 12.

A. De Pablo, R. Ferreira, F. Quiros AND J.L. Vazquez, Blow - up. El problema matemático de explosión para ecuaciones y sistemas de reacción - difusión. Boletín de la Sociedad Española de Matemática Aplicada. 2005; 32: 75 - 111.

P.G. Dlamini AND M. Khumalo, On the Computation of blow up solutions for the semilinear ODEs and

parabolic PDEs. Hindawi Publishing Corporation Mathematical Problems in Engineering. 2011; 2012.

R. Ferreira, A. De Pablo, M. Perez - Llanos AND J.D. Rossi, Critical exponents for a semilinear parabolic

equations with variable reaction.

V.A. Galaktionov AND J.L Vazquez, The problem of blow - up in nonlinear parabolic equations. Discrete

and Continous Dynamical Systems.Abr 2002; 8(2): 399 - 433.

V.A. Galaktionov AND J.L. Vazquez,Continuation of blow - up solutions of nonlinear heat equations in

several space dimensions. Communications on Pure and Applied Mathematics. 1997; 1: 1 - 67.

A. Gmira AND L. Veron, Large time behaviour of the solutions of a semilinear parabolic equation en Rn.

Journal of Differential Equations 1984. 53: 258 - 276.

F.D. Goodwill, Numerical simulation of finite - time blow - up in nonlinear ODEs, reaction - diffusion and

VIDEs [Tesis de Maestr´ıa]. University of Johannesburg. Facultad de Ciencias. 2012.

P. Grindrod, The theory and applications of reaction - diffution equations patterns and waves. 2a ed. Oxford:

Clarendon Press; 1996.

E.K. Gustafon, Introduction to Partial Differential Equations and Hilbert Space Methods. Dover Publications

INC. 3a ed. New York: DoverPublications; 1999.

B. Hu AND H.M. Yin, The Profile Near Blowup Time for Solution of the Heat Equation with a Nonlinear

Boundary Condition IMA Preprint Series Nro. 1116. Mar 1993.

G.M. Iancu AND M.W. Wong, Global solutions of semilinear heat equations in Hilbert Spaces. 1996.

N. Ioku, The Cauchy problem for heat equation with exponential nonlinearity. Journal of Differential Equations

251: 1172 - 1194.

K. Ishige, N. Mizoguchi AND H. Yagisita, Blow - up profile for nonlinear heat equation with the neumann

boundary condition. oct 2003. Artículo Electrónico.

A. Kohda AND T. Suzuki, Blow up criteria for semilinear parabolic equations.

C. Kuttler, Reaction - Difussion Equations with Applications.

V. Lakshmikantham, Comparison results for reaction diffusion equations in a Banach Space. Lecture Notes.

M. Loayza, The heat equation with singular nonlinearity and singular initial data.

L. Lorenzi, A. Lunardi, G. Metafune AND D. Pallara, Analytic Semigroups and Reaction - Diffusion

Problems. Internet Seminar. 2004 - 2005.

R.N. Machado, Una ecuaci´on no lineal de calor con valor inicial singular [Tesis de Maestr´ıa]. Universidad

Federal de Pernambuco - Centro de Ciencias Exactas de la Naturaleza; 2009.

R. Meneses AND A. Quaas, Existence and non - existence of global solutions for uniformly parabolic equations.

R. Meneses AND A. Quaas, Fujita type exponent for fully nonlinear parabolic equations ans existence results.

A. Mounmeni AND L.S.Derradji, Global Existence of Solutions for Reaction Diffusion Systems. IAENG

International Journal of Applied Mathematics. May 2010; 40(2).

A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer - New York. 1983.

M.T. Perez, Formaci´on de singularidades en algunos problemas de reacci´on - difusi´on no lineales [Tesis de Doctorado]. Departamento de Matem´aticas. Universidad Carlos III de Madrid. 2007.

R. Pinsky, Positive solutions of reaction diffusion equations with superlinear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationay solutions. Technion - Israel Institute of Technology. 1991.

A. Pulkkinen, Some comments concerning the blow - up of solutions of the exponential reaction - diffusion

equation. MATH AP. Feb 2011.

A. Pulkkinen, Blow - up in reaction - diffusion equations with exponential and power - type nonlinearieties. Aalto University School of Science. Disertación Doctoral. Jun 2011.

F. Quiros , J.D. Rossi AND J.L Vazquez, Complete Blow - Up and Thermal Avalanche for Heat Equations

With Nonlinear Boundary Conditions. Communications in Partial Differential Equations. 2002; 27: 395 - 424.

P. Quittner, P. Souplet AND M. Winkler, Initial blow up rates and universal bounds for nonlinear heat

equations. Journal of Differential Equations. 2004; 196: 316 - 339.

M.A. Rincon, J. Lmaco AND I. Liu, Existence and uniqueness of solutions of a nonlinear heat equation. T.

Mathematical Applications Computacional. 2005; 6(2): 273 - 284.

J. Smoller, Shock Waves and Reaction - Diffusion Equations. Springer - Verlag, New York. 1983.

V. Volpert AND S. Petrovskii, Reaction - diffusion waves in biology. Physics of Life Reviews. 2009: 6; 267

- 310.

J.L. Vazquez, The problems of blow up for nonlinear heat equations. Complete blow up and avalanche formation. 2004.

L. Yacheng, X. Runzhang AND Y. Tao, Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations. Nonlinear Analysis 2008; 68: 3332 - 3348.

Received: Jan. 20, 2017.

Accepted: May. 20, 2017.

Corresponding author: jjbecerras@ucvvirtual.edu.pe




DOI: http://dx.doi.org/10.17268/sel.mat.2017.01.09

Enlaces refback

  • No hay ningún enlace refback.


Short Title: Sel. mat.

-------------------------------------------------------------------------------------

 ISSN:  2411-1783  Versión Electrónica.                      

-------------------------------------------------------------------------------------

Derechos reservados © 2014 Departamento de Matemáticas.

                  E-mail: selecmat@unitru.edu.pe

 

Licencia de Creative Commons 

Selecciones Matemáticas es una revista de la Universidad Nacional de Trujillo publica sus contenidos bajo licencia Creative Commons Reconocimiento-NoComercial 3.0.