Quenching profiles for one-dimensional semilinear heat equations
Authors:
Stathis Filippas and Jong-Shenq Guo
Journal:
Quart. Appl. Math. 51 (1993), 713-729
MSC:
Primary 35B40; Secondary 35K55, 35K65
DOI:
https://doi.org/10.1090/qam/1247436
MathSciNet review:
MR1247436
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Abstract: We are interested in the local behavior, near a quenching point, of a solution of a semilinear heat equation with singular powerlike absorption. Using the method of Herrero and Velazquez, we obtain a precise description of the spatial profile of the solution in a neighborhood of a quenching point at the quenching time, under certain assumptions on the initial data.
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M. Fila, J. Hulshof, and P. Quittner, The quenching problem on N-dimensional ball, Proceedings of the International Conference on Nonlinear Diffusion Equations and Their Equilibrium States (edited by N. G. Lloyd, J. Serrin, W.-M. Ni, and L. A. Peletier), Gregynog, Wales, August 1989
- Marek Fila and Bernhard Kawohl, Asymptotic analysis of quenching problems, Rocky Mountain J. Math. 22 (1992), no. 2, 563–577. MR 1180720, DOI https://doi.org/10.1216/rmjm/1181072749
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- Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), no. 2, 262–288. MR 1056055, DOI https://doi.org/10.1137/1032046
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S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390, 79–96 (1988)
J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981
K. Deng and H. A. Levine, On the blowup of ${u_t}$ at quenching, Proc. Amer. Math. Soc. 106, 1049–1056 (1989)
M. Fila and J. Hulshof, A note on the quenching rate, Proc. Amer. Math. Soc. 112, 473–477 (1991)
M. Fila, J. Hulshof, and P. Quittner, The quenching problem on N-dimensional ball, Proceedings of the International Conference on Nonlinear Diffusion Equations and Their Equilibrium States (edited by N. G. Lloyd, J. Serrin, W.-M. Ni, and L. A. Peletier), Gregynog, Wales, August 1989
M. Fila and B. Kawohl, Asymptotic analysis of quenching problems, Rocky Mt. J. Math. 22, 563–577 (1992)
S. Filippas and R. V. Kohn, Refined asymptotics for the blow up of ${u_t} - \Delta u = {u^p}$ , Comm. Pure Appl. Math. 45, 821–869 (1992)
A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34, 425–447 (1985)
Y. Giga and R. V. Kohn, Asymptoticlly self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38, 297–319 (1985)
Y. Giga and R. V. Kohn, Characterizing blow-up using similarity variables, Indiana Univ. Math. J. 36, 1–40 (1987)
Y. Giga and R. V. Kohn, Nondegeneracy of blow-up for semilinear heat equation, Comm. Pure Appl. Math. 42, 845–884 (1989)
J.-S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation, J. Math. Anal. Appl. 151, 58–79 (1990)
J.-S. Guo, On the semilinear elliptic equation $\Delta w - \frac {1}{2}y.\nabla w + \lambda w - {w^{ - \beta }} = 0$ in ${R^n}$ , Chinese J. Math. 19, 355–377 (1991)
J.-S. Guo, On the quenching rate estimate, Quart. Appl. Math. 49, 747–752 (1991)
J.-S. Guo, The critical length for a quenching problem, Nonlinear Analysis 18, 507–516 (1992)
M. A. Herrero and J. J. L. Velazquez, Blow-up behavior of one-dimensional semilinear parabolic equations, Annales Inst. Henri Poincaré, Sec. C: Analyse Nonlineaire (to appear)
M. A. Herrero and J. J. L. Velazquez, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations 5, 973–997 (1992)
M. A. Herrero and J. J. L. Velazquez, Blow-up profiles in one-dimensional semilinear parabolic problems, Comm. Partial Differential Equations 17, 205–219 (1992)
M. A. Herrero and J. J. L. Velazquez, Generic behavior of one-dimensional blowup patterns, Ann. Scuola Normale Sup. Pisa (to appear)
H. A. Levine, Quenching, nonquenching, and beyond quenching for solutions of some parabolic equations, Annali di Mat. Pura et Applicata 155, 243–260 (1990)
H. A. Levine, Advances in Quenching, Proceedings of the International Conference on Nonlinear Diffusion Equations and Their Equilibrium States (edited by N. G. Lloyd, J. Serrin, W.-M. Ni, and L. A. Peletier), Gregynog, Wales, August 1989
H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review 32, 262–288 (1990)
J. J. L. Velazquez, Local behavior near blow-up points for semilinear parabolic equations, J. Differential Equations (to appear)
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© Copyright 1993
American Mathematical Society
