SciELO - Scientific Electronic Library Online

 
vol.53 issue1The Gauss decomposition of products of spherical harmonicsSpectral Inclusions for Quasi-Fredholm and Saphar Spectra for Integrated Semigroups author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Revista Colombiana de Matemáticas

Print version ISSN 0034-7426

Rev.colomb.mat. vol.53 no.1 Bogotá Jan./June 2019

 

Original articles

Blow up and globality of solutions for a nonautonomous semilinear heat equation with Dirichlet condition

Explosión y globalidad de soluciones para una ecuación de calor semilineal no autónoma con condición de Dirichlet

MARCOS JOSÍAS CEBALLOS-LIRA1 

AROLDO PÉREZ2 

1 Universidad Juárez Autónoma de Tabasco, Cunduacán, Tabasco, México. División Académica de Ciencias Básicas Universidad Juárez Autónoma de Tabasco Cunduacán, Tabasco, México. e-mail: marjocel 81@hotmail.com

2 Universidad Juárez Autónoma de Tabasco, Cunduacán, Tabasco, México. División Académica de Ciencias Básicas Universidad Juárez Autónoma de Tabasco Cunduacán, Tabasco, México. e-mail: aroldopz2@gmail.com


ABSTRACT.

In this paper we prove the local existence of a nonnegative mild solution for a nonautonomous semilinear heat equation with Dirichlet condition, and give sufficient conditions for the globality and for the blow up in finite time of the mild solution. Our approach for the global existence goes back to the Weissler's technique and for the finite time blow up we uses the intrinsic ultracontractivity property of the semigroup generated by the diffusion operator.

Key words and phrases: Reaction-diffusion equations; finite time blow up; Levy processes; Dirichlet problem; ultracontractive semigroup; killed process

RESUMEN.

En este artículo demostramos la existencia local de una solución "mild" no negativa para una ecuación de calor semilineal no autónoma con condición de Dirichlet, y damos condiciones suficientes para la globalidad y la explosión en tiempo finito de la solución "mild". Nuestro enfoque para la existencia global se remonta a la técnica de Weissler y para la explosión en tiempo finito utilizamos la ultracontractividad intrínseca del semigrupo generado por el operador de difusión.

Palabras y frases clave: Ecuaciones de reacción-difusión; explosión en tiempo finito; procesos de Levy; problema de Dirichlet; semigrupo ultracontractivo; proceso matado

Text complete and PDF

References

[1] C. Bandle and H. Brunner, Blowup in diffusion equations: a survey, J. Comput. Appl. Math. 97 (1998), 3-22. [ Links ]

[2] J. Bebernes and D. Eberly, Mathematical problems from combustion theory, Springer-Verlag, 1989. [ Links ]

[3] M. Birkner, J. A. Lopez-Mimbela, and A. Walkonbinger, Comparison results and steady states for the fujita equation with fractional laplacian, Annales de L'Institute Henri Poncare-Analyse non Lineare 22 (2005), 83-97. [ Links ]

[4] K. Bogdan, T. Grzywny, and M. Ryznar, Dirichlet heat kernelfor unimodal levy processes, Stochastic Process. Appl. 124 (2014), 3612-3650. [ Links ]

[5] M. Bogoya, Sobre la explosión de una ecuación de difusión no local con termino de reacción, Boletín de Matemáticas 24 (2017), no. 2, 117-130. [ Links ]

[6] E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians, J. Funct. Anal. 59 (1984), 335-395. [ Links ]

[7] K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000), 45-126. [ Links ]

[8] M. Fila, H. Ninomiya, and J. L. Vázquez, Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems, Discr. Cont. Dyn. Systems 14 (2006), 63-74. [ Links ]

[9] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana. Univ. Math. J. 34 (1985), 425-447. [ Links ]

[10] Y. Fujishima, Global existence and blow-up of solutions for the heat equation with exponential nonlinearity, J. Differential Equations 264 ( 2018), 6809-6842. [ Links ]

[11] H. Fujita, On the blowing up of solutions of the cauchy problem for ut = δu + δi+α, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109-124. [ Links ]

[12] ______, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, IL, 1968), Amer. Math. Soc., Providence, R. I. (1970), 105-113. [ Links ]

[13] P. Groisman, J. D. Ross, and H. Zaag, On the dependence of the blow-up time with respect to the initial data in a semilinear parabolic problem, Commun. Partial Differ. Equations 28 (2003), 737-744. [ Links ]

[14] T. Grzywny, Intrinsic ultracontractivity for levy processes, Probab. Math. Statist. 28 (2008), 91-106. [ Links ]

[15] M. Guedda and M. Kirane, Critically for some evolution equations, Differential Equations 37 (2001), 540-550. [ Links ]

[16] Jr. J. A. Mann and W. A. Woyczynáski, Growing fractal interfaces in the presence of self-similar hopping surface diffusion, Phys. A. 291 (2001), 159-183. [ Links ]

[17] S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Commun. Pure Appl. Math. 16 (1963), 305-333. [ Links ]

[18] E. T. Kolkovska, J. A. López-Mimbela, and A. Pérez, Blow-up and life span bounds for a reaction-diffusion equation with a time-dependent generator, Elec. J. Diff. Equations 2008 (2008), 1-18. [ Links ]

[19] T. Kulczycki and M. Ryznar, Gradient estimates of harmonic functions and transition densities for levy processes, Trans. Amer. Math, Soc. 368 (2016), 281-318. [ Links ]

[20] J. A. Lopez-Mimbela and A. Perez, Finite time blow up and stability of a semilinear equation with a time dependent levy generator, Stoch. Models 22 (2006), 735-752. [ Links ]

[21] ______, Global and nonglobal solutions of a system of nonautonomous semilinear equations with ultracontractive levy generators, J. Math. Anal. Appl. 423 (2015), 720-733. [ Links ]

[22] J. A. López-Mimbela and A. Torres, Intrinsic ultracontractivity and blowup of a semilinear dirichlet boundary value problem, Aportaciones Mat., Modelos Estocasticos, Sociedad Matematica Mexicana 14 (1998), 283-290. [ Links ]

[23] V. Marino, F. Pacella, and B. Sciunzi, Blow up of solutions of semilinear heat equations in general domains, Commun. Contemp. Math. 17 (2015), no. 2. [ Links ]

[24] L. E. Payne and G. A. Philippin, Blow-up phenomena in parabolic problems with time dependent coefficients under dirichlet boundary conditions, Proc. Amer. Math. Soc. 141 (2013), 2309-2318. [ Links ]

[25] L. E. Payne and P. W. Schaefer, Lower bound for blow-up time in parabolic problems under dirichlet conditions, J. Math. Anal. Appl. 328 (2007), 1196-1205. [ Links ]

[26] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, 1983. [ Links ]

[27] A. Perez and J. Villa, A note on blow-up of a nonlinear integral equation, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 891-897. [ Links ]

[28] M. Perez-Llanos and J. D. Rossi, Blow-up for a non-local diffusion problem with neumann boundary conditions and a reaction term, Nonlinear Analysis 70 (2009), 1629-1640. [ Links ]

[29] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in quasilinear parabolic equations, The Gruyter Ex-positions in Mathematics, 19; Walter de Gruyter & Co., 1995. [ Links ]

[30] K.-I. Sato, Levy processes and infinitely divisible distributions, Cambridge Stud. Adv. Math, vol. 68, Cambridge University Press, 1999. [ Links ]

[31] M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch, Levy flights and related topics in physics, Lecture Notes in Physics 450; Springer-Verlag, 1995. [ Links ]

[32] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math. 12 (1975), 45-51. [ Links ]

[33] V. Varlamov, Long-time asymptotics for the nonlinear heat equation with a fractional laplacian in a ball, Studia Math. 142 (2000), 71-99. [ Links ]

[34] J. Villa-Morales, An osgood condition for a semilinear reaction-diffusion equation with time-dependent generator, Arab J. Math. Sci. 22 (2016), 86-95. [ Links ]

[35] X. Wang, On the cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc. 337 (1993), 549-590. [ Links ]

[36] F. B. Weissler, Existence and nonexistence of global solutions for a semi-linear heat equation, Israel J. Math. 38 (1981), 29-40. [ Links ]

Received: June 2018; Accepted: November 2018

2010 Mathematics Subject Classification. 35K57, 35B44, 35B09, 35C15, 60G51.

This is an open-access article distributed under the terms of the Creative Commons Attribution License

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License