Global existence and stability of some semilinear problems
Kirane, Mokhtar ; Tatar, Nasser-eddine
Archivum Mathematicum, Tome 036 (2000), p. 33-44 / Harvested from Czech Digital Mathematics Library

We prove global existence and stability results for a semilinear parabolic equation, a semilinear functional equation and a semilinear integral equation using an inequality which may be viewed as a nonlinear singular version of the well known Gronwall and Bihari inequalities.

Publié le : 2000-01-01
Classification:  34D05,  34G20,  34K05,  34K20,  35B35,  35K55
@article{107716,
     author = {Mokhtar Kirane and Nasser-eddine Tatar},
     title = {Global existence and stability of some semilinear problems},
     journal = {Archivum Mathematicum},
     volume = {036},
     year = {2000},
     pages = {33-44},
     zbl = {1048.34102},
     mrnumber = {1751612},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107716}
}
Kirane, Mokhtar; Tatar, Nasser-eddine. Global existence and stability of some semilinear problems. Archivum Mathematicum, Tome 036 (2000) pp. 33-44. http://gdmltest.u-ga.fr/item/107716/

G. Butler; T. Rogers A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations, J. Math. Anal. and Appl. 33 No 1 (1971), 77–81. (1971) | MR 0270089 | Zbl 0209.42503

G. Daprato; M. Iannelli Regularity of solutions of a class of linear integrodifferential equations in Banach spaces, J. Integral Equations Appl. 8 (1985), 27–40. (1985) | MR 0771750

W. E. Fitzgibbon Semilinear functional differential equations in Banach space, J. Diff. Eq. 29 (1978), 1–14. (1978) | MR 0492663 | Zbl 0392.34041

A. Friedman Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. (1969) | MR 0445088 | Zbl 0224.35002

Y. Fujita Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math. 27 (1990), 309–321. (1990) | MR 1066629 | Zbl 0796.45010

H. Hattori; J. H. Lightbourne Global existence and blow up for a semilinear integral equation, J. Integral Equations Appl. V2, No4 (1990), 529–546. (1990) | MR 1094482

D. Henry Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin, Heidelberg, New York, 1981. (1981) | MR 0610244 | Zbl 0456.35001

H. Hoshino On the convergence properties of global solutions for some reaction-diffusion systems under Neumann boundary conditions, Diff. and Int. Eq. V9 No4 (1996), 761–778. (1996) | MR 1401436 | Zbl 0852.35023

M. Kirane; N. Tatar Asymptotic stability and blow up for a fractional evolution equation, submitted.

M. Medved’ A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, J. Math. Anal. and Appl. 214 (1997), 349–366. (1997) | MR 1475574 | Zbl 0893.26006

M. Medved’ Singular integral inequalities and stability of semilinear parabolic equations, Archivum Mathematicum (Brno) Tomus 24 (1998), 183–190. (1998) | MR 1629697 | Zbl 0915.34057

M. W. Michalski Derivatives of noninteger order and their applications, ”Dissertationes Mathematicae”, Polska Akademia Nauk, Instytut Matematyczny, Warszawa 1993. (1993) | MR 1247113 | Zbl 0880.26007

M. Miklavčič Stability for semilinear equations with noninvertible linear operator, Pacific J. Math. 1, 118 (1985), 199–214. (1985) | MR 0783024

S. M. Rankin Existence and asymptotic behavior of a functional differential equation in a Banach space, J. Math. Anal. Appl. 88 (1982), 531–542. (1982) | MR 0667076

R. Redlinger On the asymptotic behavior of a semilinear functional differential equation in Banach space, J. Math. Anal. Appl. 112 (1985), 371–377. (1985) | MR 0813604 | Zbl 0598.34053

C. Travis; G. Webb Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 395–418. (1974) | MR 0382808 | Zbl 0299.35085

C. Travis; G. Webb Existence, stability and compacteness in the $\alpha $-norm for partial functional differential equations, Trans. Amer. Math. Soc. 240 (1978), 129–143. (1978) | MR 0499583