Global existence and stability of some semilinear problems

Mokhtar Kirane; Nasser-eddine Tatar

Archivum Mathematicum (2000)

  • Volume: 036, Issue: 1, page 33-44
  • ISSN: 0044-8753

Abstract

top
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.

How to cite

top

Kirane, Mokhtar, and Tatar, Nasser-eddine. "Global existence and stability of some semilinear problems." Archivum Mathematicum 036.1 (2000): 33-44. <http://eudml.org/doc/248556>.

@article{Kirane2000,
abstract = {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.},
author = {Kirane, Mokhtar, Tatar, Nasser-eddine},
journal = {Archivum Mathematicum},
keywords = {semilinear parabolic equation; functional differential equation; integrodifferential equation; integral equation fractional evolution equation; global existence; stability; variation of parameters; semilinear parabolic equation; functional-differential equation; integrodifferential equation; integral equation; fractional evolution equation; global existence; stability; variation of parameters},
language = {eng},
number = {1},
pages = {33-44},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Global existence and stability of some semilinear problems},
url = {http://eudml.org/doc/248556},
volume = {036},
year = {2000},
}

TY - JOUR
AU - Kirane, Mokhtar
AU - Tatar, Nasser-eddine
TI - Global existence and stability of some semilinear problems
JO - Archivum Mathematicum
PY - 2000
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 036
IS - 1
SP - 33
EP - 44
AB - 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.
LA - eng
KW - semilinear parabolic equation; functional differential equation; integrodifferential equation; integral equation fractional evolution equation; global existence; stability; variation of parameters; semilinear parabolic equation; functional-differential equation; integrodifferential equation; integral equation; fractional evolution equation; global existence; stability; variation of parameters
UR - http://eudml.org/doc/248556
ER -

References

top
  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. (1971) Zbl0209.42503MR0270089
  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. (1985) MR0771750
  3. W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Diff. Eq. 29 (1978), 1–14. (1978) Zbl0392.34041MR0492663
  4. A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. (1969) Zbl0224.35002MR0445088
  5. Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math. 27 (1990), 309–321. (1990) Zbl0796.45010MR1066629
  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. (1990) MR1094482
  7. D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin, Heidelberg, New York, 1981. (1981) Zbl0456.35001MR0610244
  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. (1996) Zbl0852.35023MR1401436
  9. M. Kirane, N. Tatar, Asymptotic stability and blow up for a fractional evolution equation, submitted. 
  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. (1997) Zbl0893.26006MR1475574
  11. M. Medved’, Singular integral inequalities and stability of semilinear parabolic equations, Archivum Mathematicum (Brno) Tomus 24 (1998), 183–190. (1998) Zbl0915.34057MR1629697
  12. M. W. Michalski, Derivatives of noninteger order and their applications, ”Dissertationes Mathematicae”, Polska Akademia Nauk, Instytut Matematyczny, Warszawa 1993. (1993) Zbl0880.26007MR1247113
  13. M. Miklavčič, Stability for semilinear equations with noninvertible linear operator, Pacific J. Math. 1, 118 (1985), 199–214. (1985) MR0783024
  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. (1982) MR0667076
  15. R. Redlinger, On the asymptotic behavior of a semilinear functional differential equation in Banach space, J. Math. Anal. Appl. 112 (1985), 371–377. (1985) Zbl0598.34053MR0813604
  16. C. Travis, G. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 395–418. (1974) Zbl0299.35085MR0382808
  17. C. Travis, G. Webb, Existence, stability and compacteness in the α -norm for partial functional differential equations, Trans. Amer. Math. Soc. 240 (1978), 129–143. (1978) MR0499583

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.