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Blowup of solutions to a porous medium equation with nonlocal boundary condition. (English) Zbl 1193.35097

Summary: We investigate the blowup properties of the positive solutions to a porous medium equation with nonlocal boundary. We obtain the blowup condition and its blowup rate estimate.

MSC:

35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
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[1] Anderson, J. R., Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Part. Diff. Eq., 16, 105-143 (1991) · Zbl 0738.35033
[2] Carlson, D. E., Linear thermoelasticity, (Encyclopedia of Physics, vol. vIa/2 (1972), Springer: Springer Berlin) · Zbl 0221.73013
[3] Carl, S.; Lakshmikantham, V., Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions, J. Math. Anal. Appl., 271, 182-205 (2002) · Zbl 1010.65041
[4] Day, W. A., Extensions of property of heat equation to linear thermoelasticity and other theories, Quart. Appl. Math., 40, 319-330 (1982) · Zbl 0502.73007
[5] Day, W. A., A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math., 40, 468-475 (1983) · Zbl 0514.35038
[6] Deng, K., Comparison Principle for some nonlocal problems, Quart. Appl. Math., 50, 517-522 (1992) · Zbl 0777.35006
[7] Deng, K.; Levine, H. A., The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl., 243, 85-126 (2000) · Zbl 0942.35025
[8] Dzhuraev, T. D.; Takhirov, J. O., A problem with nonlocal boundary conditions for a quasilinear parabolic equation, Georgian Math. J., 6, 421-428 (1999) · Zbl 0938.35079
[9] Friedman, A.; Mcleod, B., Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-447 (1985) · Zbl 0576.35068
[10] Friedman, A., Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math., 44, 3, 401-407 (1986) · Zbl 0631.35041
[11] Levine, H. A., The role of critical exponents in blow-up theorems, SIAM Rev., 32, 262-288 (1990) · Zbl 0706.35008
[12] Lin, Z. G.; Lin, Y. R., Uniform blowup profiles for diffusion equations with nonlocal source and nonlocal boundary, Acta Math. Sci., 24B, 3, 443-450 (2004) · Zbl 1065.35150
[13] Pao, C. V., Dynamics of reaction-diffusion equations with nonlocal boundary conditions, Quart. Appl. Math., 50, 173-186 (1995) · Zbl 0822.35070
[14] Pao, C. V., Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math., 88, 225-238 (1998) · Zbl 0920.35030
[15] Pao, C. V., Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math., 136, 227-243 (2001) · Zbl 0993.65094
[16] Seo, S., Blowup of solutions to heat equations with nonlocal boundary conditions, Kobe J. Math., 13, 123-132 (1996) · Zbl 0874.35018
[17] Seo, S., Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions, Pacific J. Math., 193, 1, 219-226 (2000) · Zbl 1092.35511
[18] Yin, H. M., On a class of parabolic equations with nonlocal boundary conditions, J. Math. Anal. Appl., 294, 712-728 (2004) · Zbl 1060.35057
[19] Yin, Y. F., On nonlinear parabolic equations with nonlocal boundary condition, J. Math. Anal. Appl., 185, 54-60 (1994)
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