On the question of global existence for reaction-diffusion systems with mixed boundary conditions

Author:
Selwyn L. Hollis

Journal:
Quart. Appl. Math. **51** (1993), 241-250

MSC:
Primary 35K57; Secondary 35B35, 35K55

DOI:
https://doi.org/10.1090/qam/1218366

MathSciNet review:
MR1218366

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Abstract: The question of global existence for solutions of reaction-diffusion systems presents fundamental difficulties in the case in which some components of the system satisfy Neumann boundary conditions while others satisfy nonhomogeneous Dirichlet boundary conditions. We discuss particular examples for which classical solutions are known to exist globally when all components satisfy the same type of boundary condition and yet either finite-time blowup occurs or else global existence is unknown when mixed boundary condition types are imposed on the system. Some positive results are presented concerning global existence in the presence of mixed boundary conditions if certain structure requirements are placed on the system, and these results are applied to some particular chemical reaction models.

**[1]**J. Bebernes and A. Lacey,*Finite time blowup for semilinear reactive-diffusive systems*, J. Differential Equations**95**, 105-129 (1992) MR**1142278****[2]**A. Haraux and A. Youkana,*On a result of K. Masuda concerning reaction-diffusion equations*, Tohoku Math. J.**40**, 159-163 (1988)**[3]**S. Hollis,*Globally bounded solutions of reaction-diffusion systems*, Ph.D. Thesis, North Carolina State Univ., 1986**[4]**S. Hollis, R. Martin, and M. Pierre,*Global existence and boundedness in reaction-diffusion systems*, SIAM J. Math. Anal.**18**, 744-761 (1987) MR**883566****[5]**S. Hollis and J. Morgan,*Partly dissipative reaction-diffusion systems and a model of phosphorus diffusion in silicon*, J. Nonlinear Analysis--Theory, Methods and Applications, in press. MR**1181346****[6]**M. Kirane,*Global bounds and asymptotics for a system of reaction-diffusion equations*, J. Math. Anal. Appl.**138**, 328-342 (1989) MR**991027****[7]**M. Kirane, private communication, 1987**[8]**O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'ceva,*Linear and quasilinear equations of parabolic type*, Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, RI, 1968 MR**0241822****[9]**R. Martin and M. Pierre,*Nonlinear reaction-diffusion systems*, Nonlinear Equations in the Applied Sciences, edited by W. F. Ames and C. Rogers, Academic Press (to appear) MR**1132045****[10]**K. Masuda,*On the global existence and asymptotic behavior of solutions of reaction-diffusion systems*, Hokkaido Math. J.**12**, 360-370 (1982) MR**719974****[11]**J. Morgan,*Global existence for semilinear parabolic systems*, SIAM J. Math. Anal.**20**, 1128-1144 (1989) MR**1009350****[12]**J. Morgan,*Boundedness and decay results for reaction-diffusion systems*, SIAM J. Math. Anal.**21**, 1172-1189 (1990) MR**1062398****[13]**W. B. Richardson and B. J. Mulvaney,*Plateau and kink in P profiles diffused into Si: A result of strong bimolecular recombination*?, Appl. Phys. Lett.**53**, 1917-1919 (1988)**[14]**M. Singh, K. Khetarpal, and M. Sharan,*A theoretical model for studying the rate of oxygenation of blood in pulmonary capillaries*, J. Math. Biol.**9**, 305-330 (1980)

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Additional Information

DOI:
https://doi.org/10.1090/qam/1218366

Article copyright:
© Copyright 1993
American Mathematical Society