A priori bounds for solutions of parabolic problems and applications

Pavol Quittner

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 2, page 329-341
  • ISSN: 0862-7959

Abstract

top
We review some recent results concerning a priori bounds for solutions of superlinear parabolic problems and their applications.

How to cite

top

Quittner, Pavol. "A priori bounds for solutions of parabolic problems and applications." Mathematica Bohemica 127.2 (2002): 329-341. <http://eudml.org/doc/249040>.

@article{Quittner2002,
abstract = {We review some recent results concerning a priori bounds for solutions of superlinear parabolic problems and their applications.},
author = {Quittner, Pavol},
journal = {Mathematica Bohemica},
keywords = {a priori estimate; blow-up rate; periodic solution; multiplicity; superlinear parabolic problems; blow-up rate; periodic solution; multiplicity},
language = {eng},
number = {2},
pages = {329-341},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A priori bounds for solutions of parabolic problems and applications},
url = {http://eudml.org/doc/249040},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Quittner, Pavol
TI - A priori bounds for solutions of parabolic problems and applications
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 2
SP - 329
EP - 341
AB - We review some recent results concerning a priori bounds for solutions of superlinear parabolic problems and their applications.
LA - eng
KW - a priori estimate; blow-up rate; periodic solution; multiplicity; superlinear parabolic problems; blow-up rate; periodic solution; multiplicity
UR - http://eudml.org/doc/249040
ER -

References

top
  1. Elliptic boundary value problems involving measures: existence, regularity, and multiplicity, Adv. Differ. Equ. 3 (1998), 753–813. (1998) MR1659273
  2. Complete blow-up after T max for the solution of a semilinear heat equation, J. Funct. Anal. 71 (1987), 142–174. (1987) MR0879705
  3. Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, Analyse Non Linéaire, Ann. Inst. H. Poincaré 2 (1985), 185–212. (1985) Zbl0599.35073MR0797270
  4. Initial blow-up for the solutions of a semilinear parabolic equation with source term, Equations aux dérivées partielles et applications, articles dédiés à Jacques-Louis Lions, Gauthier-Villars, Paris, 1998, pp. 189–198. (1998) MR1648222
  5. On a class of superlinear elliptic problems, Commun. Partial Differ. Equations 2 (1977), 601–614. (1977) MR0509489
  6. Solutions globales d’équations de la chaleur semi linéaires, Commun. Partial Differ. Equations 9 (1984), 955–978. (1984) MR0755928
  7. Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comen. 60 (1991), 35–103. (1991) MR1120596
  8. On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc. 293 (1986), 171–189. (1986) Zbl0619.35058MR0814919
  9. A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proc. Amer. Math. Soc. 102 (1988), 131–136. (1988) Zbl0653.35039MR0915730
  10. Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view, Math. Ann. 317 (2000), 347–387. (2000) MR1764243
  11. A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1982), 41–63. (1982) MR0664341
  12. Boundedness of global solutions of nonlinear parabolic problems, Proc. of the 4th European Conf. on Elliptic and Parabolic Problems, Rolduc 2001, to appear. (to appear) 
  13. Global solutions of a semilinear parabolic equation, Adv. Differ. Equ. 4 (1999), 163–196. (1999) MR1674359
  14. Linear and nonlinear heat equations in L δ q spaces and universal bounds for global solutions, Math. Ann. 320 (2001), 87–113. (2001) MR1835063
  15. Fast blow-up mechanism for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity, R. Soc. Lond. Proc. Ser. A 456 (2000), 2957–2982. (2000) MR1843848
  16. Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Commun. Pure Applied Math. 50 (1997), 1–67. (1997) 
  17. A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equations 6 (1981), 883–901. (1981) MR0619749
  18. A bound for global solutions of semilinear heat equations, Commun. Math. Phys. 103 (1986), 415–421. (1986) Zbl0595.35057MR0832917
  19. Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1–40. (1987) MR0876989
  20. Explosion de solutions d’équations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 141–145. (1994) MR1288393
  21. A blow up result for semilinear heat equations in the supercritical case, Preprint. 
  22. Periodic solutions in superlinear parabolic problems, Acta Math. Univ. Comen (to appear). (to appear) MR1943012
  23. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form P u t = - A u + F ( u ) , Arch. Rational Mech. Anal. 51 (1973), 371–386. (1973) MR0348216
  24. Universal blow-up estimates and decay rates for a semilinear heat equation, Preprint. 
  25. On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differ. Equations 54 (1984), 97–120. (1984) MR0756548
  26. Positive solutions of nonlinear elliptic boundary value problems, J. Math. Anal. Appl. 51 (1975), 461–482. (1975) Zbl0304.35047MR0382850
  27. Eigenfunctions of the equation Δ u + λ f ( u ) = 0 , Soviet Math. Dokl. 5 (1965), 1408–1411. (1965) MR0192184
  28. A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comen. 68 (1999), 195–203. (1999) Zbl0940.35112MR1757788
  29. A priori estimates of global solutions and multiple equilibria of a superlinear parabolic problem involving measure, Electronic J. Differ. Equations 2001 (2001), no. 29, 1–17. (2001) MR1836797
  30. Universal bound for global positive solutions of a superlinear parabolic problem, Math. Ann. 320 (2001), 299–305. (2001) Zbl0981.35010MR1839765
  31. Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math (to appear). (to appear) Zbl1034.35013MR1998164
  32. Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, NoDEA, Nonlinear Differ. Equations Appl (to appear). (to appear) Zbl1058.35120MR2210288
  33. A priori estimates of global solutions of superlinear parabolic problems without variational structure, Discrete Contin. Dyn. Systems (to appear). (to appear) MR1974428
  34. Bounds of solutions of parabolic problems with nonlinear boundary conditions, In preparation. 
  35. Initial blow-up rates and universal bounds for nonlinear heat equations, Preprint. MR2028111
  36. A priori bounds for positive solutions of nonlinear elliptic equations in two variables, Duke Math. J. 41 (1974), 759–774. (1974) Zbl0294.35033MR0364859
  37. A remark on the energy blow-up behavior for nonlinear heat equations, Duke Math. J. 103 (2000), 545–556. (2000) Zbl0971.35042MR1763658

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.