Title:
|
Global existence and stability of some semilinear problems (English) |
Author:
|
Kirane, M. |
Author:
|
Tatar, N.-E. |
Language:
|
English |
Journal:
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Archivum Mathematicum |
ISSN:
|
0044-8753 (print) |
ISSN:
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1212-5059 (online) |
Volume:
|
36 |
Issue:
|
1 |
Year:
|
2000 |
Pages:
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33-44 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
Summary:
|
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. (English) |
Keyword:
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semilinear parabolic equation |
Keyword:
|
functional differential equation |
Keyword:
|
integrodifferential equation |
Keyword:
|
integral equation fractional evolution equation |
Keyword:
|
global existence |
Keyword:
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stability |
Keyword:
|
variation of parameters |
MSC:
|
34D05 |
MSC:
|
34G20 |
MSC:
|
34K05 |
MSC:
|
34K20 |
MSC:
|
35B35 |
MSC:
|
35K55 |
idZBL:
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Zbl 1048.34102 |
idMR:
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MR1751612 |
. |
Date available:
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2008-06-06T22:25:08Z |
Last updated:
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2012-05-10 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/107716 |
. |
Reference:
|
[1] 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. Zbl 0209.42503, MR 0270089 |
Reference:
|
[2] 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. MR 0771750 |
Reference:
|
[3] W. E. Fitzgibbon: Semilinear functional differential equations in Banach space.J. Diff. Eq. 29 (1978), 1–14. Zbl 0392.34041, MR 0492663 |
Reference:
|
[4] A. Friedman: Partial Differential Equations.Holt, Rinehart and Winston, New York, 1969. Zbl 0224.35002, MR 0445088 |
Reference:
|
[5] Y. Fujita: Integrodifferential equation which interpolates the heat equation and the wave equation.Osaka J. Math. 27 (1990), 309–321. Zbl 0796.45010, MR 1066629 |
Reference:
|
[6] H. Hattori, J. H. Lightbourne: Global existence and blow up for a semilinear integral equation.J. Integral Equations Appl. V2, No4 (1990), 529–546. MR 1094482 |
Reference:
|
[7] D. Henry: Geometric theory of semilinear parabolic equations.Springer-Verlag, Berlin, Heidelberg, New York, 1981. Zbl 0456.35001, MR 0610244 |
Reference:
|
[8] 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. Zbl 0852.35023, MR 1401436 |
Reference:
|
[9] M. Kirane, N. Tatar: Asymptotic stability and blow up for a fractional evolution equation.submitted. |
Reference:
|
[10] 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. Zbl 0893.26006, MR 1475574 |
Reference:
|
[11] M. Medved’: Singular integral inequalities and stability of semilinear parabolic equations.Archivum Mathematicum (Brno) Tomus 24 (1998), 183–190. Zbl 0915.34057, MR 1629697 |
Reference:
|
[12] M. W. Michalski: Derivatives of noninteger order and their applications.”Dissertationes Mathematicae”, Polska Akademia Nauk, Instytut Matematyczny, Warszawa 1993. Zbl 0880.26007, MR 1247113 |
Reference:
|
[13] M. Miklavčič: Stability for semilinear equations with noninvertible linear operator.Pacific J. Math. 1, 118 (1985), 199–214. MR 0783024 |
Reference:
|
[14] S. M. Rankin: Existence and asymptotic behavior of a functional differential equation in a Banach space.J. Math. Anal. Appl. 88 (1982), 531–542. MR 0667076 |
Reference:
|
[15] R. Redlinger: On the asymptotic behavior of a semilinear functional differential equation in Banach space.J. Math. Anal. Appl. 112 (1985), 371–377. Zbl 0598.34053, MR 0813604 |
Reference:
|
[16] C. Travis, G. Webb: Existence and stability for partial functional differential equations.Trans. Amer. Math. Soc. 200 (1974), 395–418. Zbl 0299.35085, MR 0382808 |
Reference:
|
[17] 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. MR 0499583 |
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