Comparison results and steady states for the Fujita equation with fractional laplacian

Birkner, Matthias; López-Mimbela, José Alfredo; Wakolbinger, Anton

doi:10.1016/j.anihpc.2004.05.002

Birkner, Matthias ; López-Mimbela, José Alfredo ; Wakolbinger, Anton

Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 1, pp. 83-97.

MR Zbl | 2 citations dans Numdam

DOI : 10.1016/j.anihpc.2004.05.002

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     title = {Comparison results and steady states for the {Fujita} equation with fractional laplacian},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {83--97},
     publisher = {Elsevier},
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Birkner, Matthias; López-Mimbela, José Alfredo; Wakolbinger, Anton. Comparison results and steady states for the Fujita equation with fractional laplacian. Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 1, pp. 83-97. doi : 10.1016/j.anihpc.2004.05.002. http://www.numdam.org/articles/10.1016/j.anihpc.2004.05.002/

Bibliographie
Cité par

[1] Abramowitz M., Stegun I.A., Handbook of Mathematical Functions, Dover, New York, 1972.

[2] Alexandrov A.D., Uniqueness theorems for surfaces in the large I-V, Vestnik Leningrad Univ. 11 (19) (1956) 5-17, 12 (7) (1957) 15-44; 13 (7) (1958) 14-26; 13 (13) (1958) 27-34; 13 (19) (1958) 5-8, English transl. in, Am. Math. Soc. Transl. 21 (1962) 341-354, 354-388, 389-403, 403-411, 412-416. | MR

[3] Bianchi G., Non-existence of positive solutions to semilinear elliptic equations on $R^{n}$ or $R_{+}^{n}$ through the method of moving planes, Comm. Partial Differential Equations 22 (9-10) (1997) 1671-1690. | MR | Zbl

[4] Birkner M., López-Mimbela J.A., Wakolbinger A., Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach, Proc. Amer. Math. Soc. 130 (8) (2002) 2431-2442. | MR | Zbl

[5] Blumenthal R.M., Getoor R.K., Markov Processes and Potential Theory, Academic Press, New York, 1968. | MR | Zbl

[6] Blumenthal R.M., Getoor R.K., Ray D.B., On the distribution of first hits for the symmetric stable processes, Trans. Amer. Math. Soc. 99 (1961) 540-554. | MR | Zbl

[7] Bogdan K., Byczkowski T., Potential theory of Schrödinger operator based on fractional Laplacian, Probab. Math. Statist. 20 (2) (2000) 293-335. | MR | Zbl

[8] Chen W.X., Li C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (3) (1991) 615-622. | MR | Zbl

[9] Feireisl E., Petzeltová H., Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Differential Integral Equations 10 (1) (1997) 181-196. | MR | Zbl

[10] Folland G.B., Introduction to Partial Differential Equations, Princeton University Press, Princeton, NJ, 1995. | MR | Zbl

[11] Fujita H., On the blowing up of solutions of the Cauchy problem for $u_{t} = Δ u + u^{1 + α}$ , J. Fac. Univ. Tokyo Sect. I 13 (1966) 109-124. | MR | Zbl

[12] Getoor R.K., Markov operators and their associated semigroups, Pacific J. Math. 9 (1959) 449-472. | MR | Zbl

[13] Gidas B., Ni W.M., Nirenberg L., Symmetry of positive solutions of nonlinear elliptic equations in $R^{n}$ , Math. Anal. Appl. Part A, Adv. Math. Suppl. Stud. 7 (1981) 369-402. | MR | Zbl

[14] Gui Ch., Ni W.-M., Wang W., On the stability and instability of positive steady states of a semilinear heat equations in $R^{n}$ , Comm. Pure Appl. Math. XLV (1992) 1153-1181. | MR | Zbl

[15] Gidas B., Spruck J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (4) (1981) 525-598. | MR | Zbl

[16] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1998. | Zbl

[17] Kobayashi K., Sirao T., Tanaka H., On the growing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977) 407-424. | MR | Zbl

[18] López-Mimbela J.A., Wakolbinger A., A probabilistic proof of non-explosion of a non-linear PDE system, J. Appl. Probab. 37 (3) (2000) 635-641. | MR | Zbl

[19] Nagasawa M., Sirao T., Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation, Trans. Amer. Math. Soc. 139 (1969) 301-310. | MR | Zbl

[20] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. | MR | Zbl

[21] Pohožaev S.I., Eigenfunctions of the equation $Δ u + λ f (u) = 0$ , Soviet Math. Dokl. 165 (1) (1965) 1408-1411. | MR | Zbl

[22] Serrin J., A symmetry problem in potential theory, Arch. Rat. Mech. Anal. 43 (1971) 304-318. | MR | Zbl

[23] Sugitani S., On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math. 12 (1975) 45-51. | MR | Zbl

[24] Wang X., On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc. 337 (1993) 549-590. | MR | Zbl

[25] Watson G.N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944. | JFM | MR

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