Skip to main content Accessibility help
×
Home

On the continuation of solutions of non-autonomous semilinear parabolic problems

  • Alexandre N. Carvalho (a1), Jan W. Cholewa (a2) and Marcelo J. D. Nascimento (a3)

Abstract

We study non-autonomous parabolic equations with critical exponents in a scale of Banach spaces Eσ, σ ∈ [0,1 + μ). We consider a suitable E1+ε-solution and describe continuation properties of the solution. This concerns both a situation when the solution can be continued as an E1+ε-solution and a situation when the E1+ε-norm of the solution blows up, in which case a piecewise E1+ε-solution is constructed.

Copyright

References

Hide All
1. Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Commun. Pure Appl. Math. 12 (1959), 623727.
2. Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II, Commun. Pure Appl. Math. 7 (1964), 3592.
3. Amann, H., Existence and regularity for semilinear parabolic evolution equations, Annali Scuola Norm. Sup. Pisa 11 (1984), 693–676.
4. Amann, H., Global existence for semilinear parabolic systems, J. Reine Angew. Math. 360 (1985), 4783.
5. Amann, H., On abstract parabolic fundamental solution, J. Math. Soc. Jpn 39 (1987), 93116.
6. Amann, H., Parabolic evolution equations in interpolation and extrapolation spaces, J. Funct. Analysis 78 (1988), 233270.
7. Amann, H., Linear and quasilinear parabolic problems, volume I: abstract linear theory, Monographs in Mathematics, Volume 89 (Birkhäuser, 1995).
8. Amann, H., Hieber, M. and Simonett, G., Bounded H-calculus for elliptic operators, Diff. Integ. Eqns 3 (1994), 613653.
9. Arrieta, J. M. and Carvalho, A. N., Abstract parabolic problems with critical nonlinearities and applications to Navier–Stokes and heat equations, Trans. Am. Math. Soc. 352 (2000), 285310.
10. Barbu, V., Nonlinear semigroups and differential equations in Banach spaces (Noordhoff, Groningen, 1976).
11. Carvalho, A. N. and Cholewa, J. W., Local well-posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc. 66 (2002), 443463.
12. Carvalho, A. N. and Cholewa, J. W., Attractors for strongly damped wave equations with critical nonlinearities, Pac. J. Math. 207 (2002), 287310.
13. Carvalho, A. N. and Cholewa, J. W., Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Analysis Applic. 310 (2005), 557578.
14. Carvalho, A. N. and Cholewa, J. W., Strongly damped wave equations in , Discrete Contin. Dynam. Syst. (Suppl.) (2007), 230239.
15. Carvalho, A. N. and Nascimento, M. J. D., Singularly non-autonomous semilinear parabolic problems with critical exponents and applications, Discrete Contin. Dynam. Syst. S 2 (2009), 449471.
16. Chen, S. and Triggiani, R., Proof of extension of two conjectures on structural damping for elastic systems: the case ½ ⩽ α ⩽ 1, Pac. J. Math. 136 (1989), 1555.
17. Cholewa, J. W. and Dlotko, T., Global attractors in abstract parabolic problems (Cambridge University Press, 2000).
18. Cholewa, J. W. and Rodriguez-Bernal, A., Linear and semilinear higher order parabolic equations in , Nonlin. Analysis TMA 75 (2012), 194210.
19. Denk, R., Dore, G., Hieber, M., Prüss, J. and Venni, A., New thoughts on old results of R. T. Seeley, Math. Annalen 328 (2004), 545583.
20. Friedman, A., Partial differential equations of parabolic type (Prentice Hall, Englewood Cliffs, NJ, 1964).
21. Henry, D., Geometric theory of semilinear parabolic equations (Springer, 1981).
22. Lunardi, A., Analytic semigroup and optimal regularity in parabolic problems (Birkhäuser, 1995).
23. Pazy, A., Semigroups of linear operators and applications to partial differential equations (Springer, 1983).
24. Prüss, J. and Sohr, H., Imaginary powers of elliptic second order differential operators in Lp-spaces, Hiroshima Math. J. 23 (1993), 161192.
25. Seeley, R., Interpolation in Lp with boundary conditions, Studia Math. 44 (1972), 4760.
26. Sobolevskiǐ, P. E., Equations of parabolic type in a Banach space, Am. Math. Soc. Transl. 2 49 (1966), 162.
27. Tanabe, H., Functional analytic methods for partial differential equations (Dekker, New York, 1997).
28. Triebel, H., Interpolation theory, function spaces, differential operators (North-Holland, Amsterdam, 1978).
29. von Wahl, W., Global solutions to evolution equations of parabolic type, in Differential equations in Banach spaces (ed. Favini, A. and Obrecht, E.), Lecture Notes in Mathematics, Volume 1223, pp. 254266 (Springer, 1986).
30. Yagi, A., Abstract parabolic evolution equations and their applications, Springer Monographs in Mathematics (Springer, 2010).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Related content

Powered by UNSILO

On the continuation of solutions of non-autonomous semilinear parabolic problems

  • Alexandre N. Carvalho (a1), Jan W. Cholewa (a2) and Marcelo J. D. Nascimento (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.