Nonlinear diffusion induced by nonlinear sources
Authors:
D. D. Joseph and E. M. Sparrow
Journal:
Quart. Appl. Math. 28 (1970), 327-342
MSC:
Primary 80.35
DOI:
https://doi.org/10.1090/qam/272272
MathSciNet review:
272272
Full-text PDF Free Access
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Additional Information
- Daniel D. Joseph, Bounds on $\lambda $ for positive solutions of $\Delta \psi +\lambda f(r)\{\psi +G(\psi )\}=0$, Quart. Appl. Math. 23 (1965/66), 349–354. MR 194732, DOI https://doi.org/10.1090/S0033-569X-1966-0194732-1
M. Jakob, Heat transfer, Vol. 1, Chap. 10, Wiley, New York, 1949
P. J. Schneider, Conduction heat transfer, Chap. 8, Addison-Wesley, Reading, Mass., 1955
D. D. Joseph, Non-linear heat generation and stability of the temperature distribution in conducting solids, Int. J. Heat Mass Transfer 8, 281–288 (1965)
P. J. Torvic, Temperature rise and stresses due to internal heating, preprint 68-HT-37, Amer. Soc. Mech. Engrs., 1968
D. Luss and N. R. Amundson, Uniqueness of the steady state solutions for chemical reaction occurring in a catalyst particle or in a tubular reactor with axial diffusion, Chem. Engng. Sci. 22, 253–266 (1967)
D. Luss, Sufficient conditions for uniqueness of the steady state solutions in distributed parameter systems, Chem. Engng. Sci. 23, 1249–1255 (1968)
D. Luss, On the uniqueness of a large distributed parameter system with chemical reaction and heat and mass diffusion, Chem. Engng. Sci. 24, 879–883 (1969)
D. Luss and J. C. M. Lee, On global stability in distributed parameter systems, Chem. Engng. Sci. 23, 1237–1248 (1968)
- I. M. Gel′fand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29 (1963), 295–381. MR 0153960
- Hiroshi Fujita, On the nonlinear equations $\Delta u+e^{u}=0$ and $\partial v/\partial t=\Delta v+e ^{v}$, Bull. Amer. Math. Soc. 75 (1969), 132–135. MR 239258, DOI https://doi.org/10.1090/S0002-9904-1969-12175-0
- S. A. Regirer, The influence of thermal effects on the viscous resistance of a steady uniform flow of liquid, J. Appl. Math. Mech. 22 (1958), 580–586 (414–418 Prikl. Mat. Meh.). MR 0102289, DOI https://doi.org/10.1016/0021-8928%2858%2990075-3
S. A. Kaganov, Establishing laminar flow for an incompressible liquid in a horizontal channel and a curved cylindrical tube with corrections for frictional heat and the temperature dependence of viscosity, Int. Chem. Engng. 3, 33–35 (1963)
R. E. Colwell and K. R. Nickolls, The screw extruder, Ind. Engng. Chem. 51, 841–843 (1959)
D. D. Joseph, Variable viscosity effects on the flow and stability of flow in channels and pipes, Phys. Fluids 7, 1761–1771 (1964)
J. Gavis and R. L. Laurence, Viscous heating in plane and circular flow between moving surfaces, I and EC Fundamentals 7, 232–239 (1968)
J. Gavis and R. L. Laurence, Viscous heating of a power-law liquid in plane flow, I and EC Fundamentals 7, 525–527 (1968)
W. K. Ergun, Self-limiting power excursions in large reactors, Trans. Amer. Nuclear Soc. 8, 221 (1965)
- Donald S. Cohen, Positive solutions of nonlinear eigenvalue problems: Applications to nonlinear reactor dynamics, Arch. Rational Mech. Anal. 26 (1967), 305–315. MR 216159, DOI https://doi.org/10.1007/BF00281661
- Donald S. Cohen, Correction to: “Positive solutions of nonlinear eigenvalue problems: Applications of nonlinear reactor dynamics”, Arch. Rational Mech. Anal. 30 (1968), 401. MR 232084, DOI https://doi.org/10.1007/BF00253671
D. Luss and N. R. Amundson, Uniqueness of the steady state for an isothermal porous catalyst, I and EC Fundamentals 6, 457–459 (1967)
- Herbert B. Keller and Donald S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361–1376. MR 0213694
- Donald S. Cohen, Positive solutions of a class of nonlinear eigenvalue problems, J. Math. Mech. 17 (1967), 209–215. MR 0213695, DOI https://doi.org/10.1512/iumj.1968.17.17010
- Herbert B. Keller, Nonexistence and uniqueness of positive solutions of nonlinear eigenvalue problems, Bull. Amer. Math. Soc. 74 (1968), 887–891. MR 229985, DOI https://doi.org/10.1090/S0002-9904-1968-12067-1
- M. M. Vaĭnberg and V. A. Trenogin, The Ljapunov and Schmidt methods in the theory of non-linear equations and their subsequent development, Uspehi Mat. Nauk 17 (1962), no. 2 (104), 13–75 (Russian). MR 0154113
D. D. Joseph, Bounds on $\lambda$ for positive solutions of $\Delta \psi + \lambda f\left ( r \right )\left \{ {\psi + G\left ( \psi \right )} \right \} = 0$, Quart. Appl. Math. 23, 349-354 (1966)
M. Jakob, Heat transfer, Vol. 1, Chap. 10, Wiley, New York, 1949
P. J. Schneider, Conduction heat transfer, Chap. 8, Addison-Wesley, Reading, Mass., 1955
D. D. Joseph, Non-linear heat generation and stability of the temperature distribution in conducting solids, Int. J. Heat Mass Transfer 8, 281–288 (1965)
P. J. Torvic, Temperature rise and stresses due to internal heating, preprint 68-HT-37, Amer. Soc. Mech. Engrs., 1968
D. Luss and N. R. Amundson, Uniqueness of the steady state solutions for chemical reaction occurring in a catalyst particle or in a tubular reactor with axial diffusion, Chem. Engng. Sci. 22, 253–266 (1967)
D. Luss, Sufficient conditions for uniqueness of the steady state solutions in distributed parameter systems, Chem. Engng. Sci. 23, 1249–1255 (1968)
D. Luss, On the uniqueness of a large distributed parameter system with chemical reaction and heat and mass diffusion, Chem. Engng. Sci. 24, 879–883 (1969)
D. Luss and J. C. M. Lee, On global stability in distributed parameter systems, Chem. Engng. Sci. 23, 1237–1248 (1968)
I. M. Gel’fand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29, 295–381 (1963),
H. Fujita, On the nonlinear equations $\Delta u + {e^u} = 0$ and $\partial \upsilon /\partial t = \Delta \upsilon + {e^\upsilon }$, Bull. Amer. Math. Soc. 75, 132–135 (1969)
S. A. Regirer, The influence of thermal effects on the viscous resistance of a steady uniform flow of liquid, J. Appl. Math. Mech. 22, 580–586 (1958)
S. A. Kaganov, Establishing laminar flow for an incompressible liquid in a horizontal channel and a curved cylindrical tube with corrections for frictional heat and the temperature dependence of viscosity, Int. Chem. Engng. 3, 33–35 (1963)
R. E. Colwell and K. R. Nickolls, The screw extruder, Ind. Engng. Chem. 51, 841–843 (1959)
D. D. Joseph, Variable viscosity effects on the flow and stability of flow in channels and pipes, Phys. Fluids 7, 1761–1771 (1964)
J. Gavis and R. L. Laurence, Viscous heating in plane and circular flow between moving surfaces, I and EC Fundamentals 7, 232–239 (1968)
J. Gavis and R. L. Laurence, Viscous heating of a power-law liquid in plane flow, I and EC Fundamentals 7, 525–527 (1968)
W. K. Ergun, Self-limiting power excursions in large reactors, Trans. Amer. Nuclear Soc. 8, 221 (1965)
D. S. Cohen, Positive solutions of nonlinear eigenvalue problems: Applications to nonlinear reactor dynamics, Arch. Rational Mech. Anal. 26, 305–315 (1967)
D. S. Cohen, Correction to reference 19, Arch. Rational Mech. Anal. 30, 401 (1968)
D. Luss and N. R. Amundson, Uniqueness of the steady state for an isothermal porous catalyst, I and EC Fundamentals 6, 457–459 (1967)
H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16, 1361–1376 (1967)
D. S. Cohen, Positive solutions of a class of nonlinear eigenvalue problems, J. Math. Mech. 17, 209–215 (1967)
H. B. Keller, Nonexistence and uniqueness of positive solutions of nonlinear eigenvalue problems, Bull. Amer. Math. Soc. 74, 887–891 (1968)
M. M. Vainberg and V. A. Trenogin, The Lyapunov and Schmidt methods in the theory of non-linear equations and their subsequent development, Russian Math. Surveys 17, no. 2, 1–60 (1962)
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© Copyright 1970
American Mathematical Society