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Year 2022, Volume: 51 Issue: 3, 757 - 774, 01.06.2022

Salim Mesbahi

https://doi.org/10.15672/hujms.936018
Cited By: 1

Abstract

References

  • [1] N. Alaa, S. Mesbahi and W. Bouarifi, Global existence of weak solutions for parabolic triangular reaction diffusion systems applied to a climate model, An. Univ. Craiova Ser. Mat. Inform. 42 (1), 80-97, 2015.
  • [2] N. Alaa, S. Mesbahi, A. Mouida and W. Bouarifi, Existence of solutions for quasilinear elliptic degenerate systems with $L^{1}$ data and nonlinearity in the gradient. Electron. J. Differential Equations 2013 (142), 1-13, 2013.
  • [3] R. Aris, Mathematical Modeling : A Chemical Engineer’s Perspective, Academic Press, New York, 1999.
  • [4] M.A. Beauregard and Q. Sheng, A fully adaptive approximation for quenching-type reaction diffusion equations over circular domains, Numerical methods in PDE, 30 (2), 472-489, 2013.
  • [5] L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37, 363-380, 2009.
  • [6] P. Constantin, A. Kiselev and L. Ryzhik, Quenching of flames by fluid advection, Comm. Pure Appl. Math. 54, 1320-1342, 2001.
  • [7] Q. Dai and Y. Gu, A Short Note on Quenching Phenomena for Semilinear Parabolic Equations, J. Differential Equations, 137, 240-250, 1997.
  • [8] A. Dall’Aglio, D. Giachetti and J.P. Puel, Nonlinear parabolic equations with natural growth in general domains, Boll. Unione Mat. Ital. Serie 8 8-B (3), 653-683, 2005.
  • [9] I. de Bonis and A. Muntean, Existence of weak solutions to a nonlinear reaction diffusion system with singular sources, Electron. J. Differential Equations 2017 (202), 1-16, 2017.
  • [10] I. de Bonis, Singular elliptic and parabolic problems : existence and regularity of solutions, Ph D Thesis, Sapienza University of Rome, 2015.
  • [11] I. de Bonis and D. Giachetti, Nonnegative solutions for a class of singular parabolic problems involving p-laplacian, Asymptot. Anal. 91, 147-183, 2015.
  • [12] I. de Bonis and L.M. De Cave, Degenerate parabolic equations with singular lower order terms, Differential Integral Equations, 27 (9/10), 949-976, 2014.
  • [13] I. de Bonis and D. Giachetti, Singular parabolic problems with possibly changing sign data, Discrete Contin. Dyn. Syst. Ser. B 7 (19), 2047-2064, 2014.
  • [14] D. Giachetti, Martinez-Aparicio and P.J., Murat, A semilinear elliptic equation with a mild singularity at $u=0$ existence and homogenization, J. Math. Pures Appl. 2016.
  • [15] H. Kawarada, On solutions of the initial boundary value problem for $u_{t}=u_{xx}+1$, RIMS Kyoto 10, 729-736, 1975.
  • [16] A. Kiselev and A. Zlatos, Quenching of combustion by shear flows, Duke Math. J. 132, 49-72, 2006.
  • [17] R. Landes and V. Mustonen, On parabolic initial-boundary value problems with critical growth for the gradient, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11, 135-158, 1994.
  • [18] R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 89, 217-237, 1981.
  • [19] O. Levenspiel, Chemical Reaction Engineering, John Wiley and Sons, New York, 1962.
  • [20] H.A. Levine, The phenomenon of quenching : A survey, Proceedings, VIth International Conference on Trends in the Theory and Practice of Nonlinear Analysis, North-Holland, New York, 1985.
  • [21] J.L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod et Gautier-Villars, Paris, 1969.
  • [22] P.T. Marion, Quenching Phenomena Due to a Concentrated Nonlinear Source in an Infinitely Long Cylinder, Journal of Applied Mathematics and Physics 7, 2015-2025, 2019.
  • [23] S. Mesbahi and N. Alaa, Mathematical analysis of a reaction diffusion model for image restoration, An. Univ. Craiova Ser. Mat. Inform. 42 (1), 70-79, 2015.
  • [24] S. Mesbahi and N. Alaa, Existence result for triangular Reaction Diffusion systems with $L^{1}$ data and critical growth with respect to the gradient, Mediterr. J. Math. 10, 255-275, 2013.
  • [25] A. Mokrane, Existence of bounded solutions of some nonlinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A 107, 313-326, 1987.
  • [26] A. Muntean, Continuum Modeling : An Approach through Practical Examples, Springer Briefs in Applied Science and Technology : Mathematical Methods, Springer, Heidelberg, 2015.
  • [27] J.D. Murray, Mathematical Biology I : An Introduction, vol. I, 3rd ed., Springer- Verlag, Berlin Heidelberg, 2003.
  • [28] J.D. Murray, Mathematical Biology II : Spatial Models and Biochemical Applications, vol. II, 3rd ed., Springer-Verlag, Berlin Heidelberg, 2003.
  • [29] M. Pierre, Global existence in reaction diffusion systems with control of mass: a survey, Milan J. Math. 78, 417-455, 2010.
  • [30] T. Salin, Quenching and blowup problems for reaction diffusion equations, Helsinki University of Technology Institute of Mathematics Research Reports A466, 2004.
  • [31] T. Salin, On quenching with logarithmic singularity, Nonlinear Anal. TMA. 52, 261- 289, 2003.
  • [32] T. Salin, Quenching-rate estimate for a reaction diffusion equation with weakly singular reaction term, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 11 (4), 2004.
  • [33] T. Salin, On a refined asymptotic analysis for the quenching problem, Helsinki University of Technology, Institute of Mathematics, Research Report A457, 2003.
  • [34] B. Selçuk, Quenching behavior of a semilinear reaction diffusion system with singular boundary condition, Turk J Math. 40, 166-180, 2016.
  • [35] J. Simon, Compact sets in the space $L^{p}(0,T,B)$, Ann. Mat. Pura Appl. 146, 65-96, 1987.
  • [36] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble,) 15, 189-258, 1965.
  • [37] Q. Wang, Quenching Phenomenon for a Parabolic MEMS Equation, Chin. Ann. Math. Ser. B, 39 (1), 129-144, 2018.

On the existence of weak solutions for a class of singular reaction diffusion systems

Year 2022, Volume: 51 Issue: 3, 757 - 774, 01.06.2022

Salim Mesbahi

https://doi.org/10.15672/hujms.936018
Cited By: 1

Abstract

We study the existence of weak solutions for a parabolic reaction diffusion model applied in Quenching endowed with singular production terms by reaction. The singularity is due to a potential occurrence of quenching localized to the domain boundary. The techniques used are based on energy estimates to approach nonsingular problems and uniform control on the set where singularities are localizing.

Keywords

singular parabolic equations weak solutions reaction diffusion systems quenching

References

  • [1] N. Alaa, S. Mesbahi and W. Bouarifi, Global existence of weak solutions for parabolic triangular reaction diffusion systems applied to a climate model, An. Univ. Craiova Ser. Mat. Inform. 42 (1), 80-97, 2015.
  • [2] N. Alaa, S. Mesbahi, A. Mouida and W. Bouarifi, Existence of solutions for quasilinear elliptic degenerate systems with $L^{1}$ data and nonlinearity in the gradient. Electron. J. Differential Equations 2013 (142), 1-13, 2013.
  • [3] R. Aris, Mathematical Modeling : A Chemical Engineer’s Perspective, Academic Press, New York, 1999.
  • [4] M.A. Beauregard and Q. Sheng, A fully adaptive approximation for quenching-type reaction diffusion equations over circular domains, Numerical methods in PDE, 30 (2), 472-489, 2013.
  • [5] L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37, 363-380, 2009.
  • [6] P. Constantin, A. Kiselev and L. Ryzhik, Quenching of flames by fluid advection, Comm. Pure Appl. Math. 54, 1320-1342, 2001.
  • [7] Q. Dai and Y. Gu, A Short Note on Quenching Phenomena for Semilinear Parabolic Equations, J. Differential Equations, 137, 240-250, 1997.
  • [8] A. Dall’Aglio, D. Giachetti and J.P. Puel, Nonlinear parabolic equations with natural growth in general domains, Boll. Unione Mat. Ital. Serie 8 8-B (3), 653-683, 2005.
  • [9] I. de Bonis and A. Muntean, Existence of weak solutions to a nonlinear reaction diffusion system with singular sources, Electron. J. Differential Equations 2017 (202), 1-16, 2017.
  • [10] I. de Bonis, Singular elliptic and parabolic problems : existence and regularity of solutions, Ph D Thesis, Sapienza University of Rome, 2015.
  • [11] I. de Bonis and D. Giachetti, Nonnegative solutions for a class of singular parabolic problems involving p-laplacian, Asymptot. Anal. 91, 147-183, 2015.
  • [12] I. de Bonis and L.M. De Cave, Degenerate parabolic equations with singular lower order terms, Differential Integral Equations, 27 (9/10), 949-976, 2014.
  • [13] I. de Bonis and D. Giachetti, Singular parabolic problems with possibly changing sign data, Discrete Contin. Dyn. Syst. Ser. B 7 (19), 2047-2064, 2014.
  • [14] D. Giachetti, Martinez-Aparicio and P.J., Murat, A semilinear elliptic equation with a mild singularity at $u=0$ existence and homogenization, J. Math. Pures Appl. 2016.
  • [15] H. Kawarada, On solutions of the initial boundary value problem for $u_{t}=u_{xx}+1$, RIMS Kyoto 10, 729-736, 1975.
  • [16] A. Kiselev and A. Zlatos, Quenching of combustion by shear flows, Duke Math. J. 132, 49-72, 2006.
  • [17] R. Landes and V. Mustonen, On parabolic initial-boundary value problems with critical growth for the gradient, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11, 135-158, 1994.
  • [18] R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 89, 217-237, 1981.
  • [19] O. Levenspiel, Chemical Reaction Engineering, John Wiley and Sons, New York, 1962.
  • [20] H.A. Levine, The phenomenon of quenching : A survey, Proceedings, VIth International Conference on Trends in the Theory and Practice of Nonlinear Analysis, North-Holland, New York, 1985.
  • [21] J.L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod et Gautier-Villars, Paris, 1969.
  • [22] P.T. Marion, Quenching Phenomena Due to a Concentrated Nonlinear Source in an Infinitely Long Cylinder, Journal of Applied Mathematics and Physics 7, 2015-2025, 2019.
  • [23] S. Mesbahi and N. Alaa, Mathematical analysis of a reaction diffusion model for image restoration, An. Univ. Craiova Ser. Mat. Inform. 42 (1), 70-79, 2015.
  • [24] S. Mesbahi and N. Alaa, Existence result for triangular Reaction Diffusion systems with $L^{1}$ data and critical growth with respect to the gradient, Mediterr. J. Math. 10, 255-275, 2013.
  • [25] A. Mokrane, Existence of bounded solutions of some nonlinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A 107, 313-326, 1987.
  • [26] A. Muntean, Continuum Modeling : An Approach through Practical Examples, Springer Briefs in Applied Science and Technology : Mathematical Methods, Springer, Heidelberg, 2015.
  • [27] J.D. Murray, Mathematical Biology I : An Introduction, vol. I, 3rd ed., Springer- Verlag, Berlin Heidelberg, 2003.
  • [28] J.D. Murray, Mathematical Biology II : Spatial Models and Biochemical Applications, vol. II, 3rd ed., Springer-Verlag, Berlin Heidelberg, 2003.
  • [29] M. Pierre, Global existence in reaction diffusion systems with control of mass: a survey, Milan J. Math. 78, 417-455, 2010.
  • [30] T. Salin, Quenching and blowup problems for reaction diffusion equations, Helsinki University of Technology Institute of Mathematics Research Reports A466, 2004.
  • [31] T. Salin, On quenching with logarithmic singularity, Nonlinear Anal. TMA. 52, 261- 289, 2003.
  • [32] T. Salin, Quenching-rate estimate for a reaction diffusion equation with weakly singular reaction term, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 11 (4), 2004.
  • [33] T. Salin, On a refined asymptotic analysis for the quenching problem, Helsinki University of Technology, Institute of Mathematics, Research Report A457, 2003.
  • [34] B. Selçuk, Quenching behavior of a semilinear reaction diffusion system with singular boundary condition, Turk J Math. 40, 166-180, 2016.
  • [35] J. Simon, Compact sets in the space $L^{p}(0,T,B)$, Ann. Mat. Pura Appl. 146, 65-96, 1987.
  • [36] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble,) 15, 189-258, 1965.
  • [37] Q. Wang, Quenching Phenomenon for a Parabolic MEMS Equation, Chin. Ann. Math. Ser. B, 39 (1), 129-144, 2018.
There are 37 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Salim Mesbahi 0000-0002-2455-3991

Publication Date June 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 3

Cite

APA Mesbahi, S. (2022). On the existence of weak solutions for a class of singular reaction diffusion systems. Hacettepe Journal of Mathematics and Statistics, 51(3), 757-774. https://doi.org/10.15672/hujms.936018
AMA Mesbahi S. On the existence of weak solutions for a class of singular reaction diffusion systems. Hacettepe Journal of Mathematics and Statistics. June 2022;51(3):757-774. doi:10.15672/hujms.936018
Chicago Mesbahi, Salim. “On the Existence of Weak Solutions for a Class of Singular Reaction Diffusion Systems”. Hacettepe Journal of Mathematics and Statistics 51, no. 3 (June 2022): 757-74. https://doi.org/10.15672/hujms.936018.
EndNote Mesbahi S (June 1, 2022) On the existence of weak solutions for a class of singular reaction diffusion systems. Hacettepe Journal of Mathematics and Statistics 51 3 757–774.
IEEE S. Mesbahi, “On the existence of weak solutions for a class of singular reaction diffusion systems”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, pp. 757–774, 2022, doi: 10.15672/hujms.936018.
ISNAD Mesbahi, Salim. “On the Existence of Weak Solutions for a Class of Singular Reaction Diffusion Systems”. Hacettepe Journal of Mathematics and Statistics 51/3 (June 2022), 757-774. https://doi.org/10.15672/hujms.936018.
JAMA Mesbahi S. On the existence of weak solutions for a class of singular reaction diffusion systems. Hacettepe Journal of Mathematics and Statistics. 2022;51:757–774.
MLA Mesbahi, Salim. “On the Existence of Weak Solutions for a Class of Singular Reaction Diffusion Systems”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, 2022, pp. 757-74, doi:10.15672/hujms.936018.
Vancouver Mesbahi S. On the existence of weak solutions for a class of singular reaction diffusion systems. Hacettepe Journal of Mathematics and Statistics. 2022;51(3):757-74.

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