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), 349β354. 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, 281β288 (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, 253β266 (1967)
D. Luss, Sufficient conditions for uniqueness of the steady state solutions in distributed parameter systems, Chem. Engng. Sci. 23, 1249β1255 (1968)
D. Luss, On the uniqueness of a large distributed parameter system with chemical reaction and heat and mass diffusion, Chem. Engng. Sci. 24, 879β883 (1969)
D. Luss and J. C. M. Lee, On global stability in distributed parameter systems, Chem. Engng. Sci. 23, 1237β1248 (1968)
- I. M. Gelβ²fand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29 (1963), 295β381. 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), 132β135. 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), 580β586 (414β418 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, 33β35 (1963)
R. E. Colwell and K. R. Nickolls, The screw extruder, Ind. Engng. Chem. 51, 841β843 (1959)
D. D. Joseph, Variable viscosity effects on the flow and stability of flow in channels and pipes, Phys. Fluids 7, 1761β1771 (1964)
J. Gavis and R. L. Laurence, Viscous heating in plane and circular flow between moving surfaces, I and EC Fundamentals 7, 232β239 (1968)
J. Gavis and R. L. Laurence, Viscous heating of a power-law liquid in plane flow, I and EC Fundamentals 7, 525β527 (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), 305β315. MR 216159, DOI https://doi.org/10.1007/BF00281661
- Donald S. Cohen, Correction to: βPositive solutions of nonlinear eigenvalue problems: Applications of nonlinear reactor dynamicsβ, 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, 457β459 (1967)
- Herbert B. Keller and Donald S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361β1376. MR 0213694
- Donald S. Cohen, Positive solutions of a class of nonlinear eigenvalue problems, J. Math. Mech. 17 (1967), 209β215. 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), 887β891. MR 229985, DOI https://doi.org/10.1090/S0002-9904-1968-12067-1
- M. M. VaΔnberg 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), 13β75 (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, 281β288 (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, 253β266 (1967)
D. Luss, Sufficient conditions for uniqueness of the steady state solutions in distributed parameter systems, Chem. Engng. Sci. 23, 1249β1255 (1968)
D. Luss, On the uniqueness of a large distributed parameter system with chemical reaction and heat and mass diffusion, Chem. Engng. Sci. 24, 879β883 (1969)
D. Luss and J. C. M. Lee, On global stability in distributed parameter systems, Chem. Engng. Sci. 23, 1237β1248 (1968)
I. M. Gelβfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29, 295β381 (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, 132β135 (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, 580β586 (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, 33β35 (1963)
R. E. Colwell and K. R. Nickolls, The screw extruder, Ind. Engng. Chem. 51, 841β843 (1959)
D. D. Joseph, Variable viscosity effects on the flow and stability of flow in channels and pipes, Phys. Fluids 7, 1761β1771 (1964)
J. Gavis and R. L. Laurence, Viscous heating in plane and circular flow between moving surfaces, I and EC Fundamentals 7, 232β239 (1968)
J. Gavis and R. L. Laurence, Viscous heating of a power-law liquid in plane flow, I and EC Fundamentals 7, 525β527 (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, 305β315 (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, 457β459 (1967)
H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16, 1361β1376 (1967)
D. S. Cohen, Positive solutions of a class of nonlinear eigenvalue problems, J. Math. Mech. 17, 209β215 (1967)
H. B. Keller, Nonexistence and uniqueness of positive solutions of nonlinear eigenvalue problems, Bull. Amer. Math. Soc. 74, 887β891 (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, 1β60 (1962)
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© Copyright 1970
American Mathematical Society
