Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-19T06:04:19.607Z Has data issue: false hasContentIssue false
  • English
  • Français

On fractional heat equations with non-local initial conditions

Part of: Parabolic equations and systems Other special functions

Published online by Cambridge University Press:  08 July 2015

Bruno de Andrade ,
Claudio Cuevas  and
Herme Soto
Bruno de Andrade
Affiliation:
Instituto de Ciências Matemáticas e de Computaҫão, Universidade de São Paulo, Campus de São Carlos, CEP 13560-970, São Carlos, São Paulo, Brazil ([email protected])
Claudio Cuevas
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco, CEP 50540-740, Recife, Pernambuco, Brazil ([email protected])
Herme Soto
Affiliation:
Departamento de Matemática e Estadística, Universidad de La Frontera, Cosilla 54D Temuco, Chile ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider the problem of existence of mild solutions to semilinear fractional heat equations with non-local initial conditions. We provide sufficient conditions for existence and regularity of such solutions.

Keywords

fractional heat equationsnon-local initial conditionsMittag-Leffler functions

MSC classification

Secondary: 35K05: Heat equation 35K55: Nonlinear parabolic equations 33E12: Mittag-Leffler functions and generalizations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 1 , February 2016 , pp. 65 - 76
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Amann, H., Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function spaces, differential operators and nonlinear analysis (ed. Schmeisser, H.-J. and Triebel, H.), Teubner-Texte zur Mathematik, Volume 133, pp. 9–126 (Teubner, Leipzig/Stuttgart, 1993).CrossRefGoogle Scholar
2. Cardinali, T. and Rubbioni, P., Impulsive mild solution for semilinear differential inclusions with nonlocal conditions in Banach spaces, Nonlin. Analysis TMA 75 (2012), 871–879.CrossRefGoogle Scholar
3. Chabrowski, J., On nonlocal problems for parabolic equations, Nagoya Math. J. 93 (1984), 109–131.Google Scholar
4. Deng, K., Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Analysis Applic. 179 (1993), 630–637.CrossRefGoogle Scholar
5. Henry, D., Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Volume 840 (Springer, 1981).Google Scholar
6. Kerefov, A. A., Non-local boundary value problems for parabolic equation, Diff. Uravn. 15 (1979), 52–55.Google Scholar
7. Kexue, L, Jigen, P. and Junxiong, J., Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives, J. Funct. Analysis 263 (2012) 476–510.Google Scholar
8. Mainardi, F., Fractional calculus and waves in linear viscoelasticity (Imperial College Press, London, 2010).Google Scholar
9. Mittag-Leffler, G. M., Une généralisation de l'integrale de Laplace–Abel, C. R. Acad. Sci. Paris II 137 (1903), 537–539.Google Scholar
10. Mittag-Leffler, G. M., Sur la nouvelle fonction E α(x), C. R. Acad. Sci. Paris II 137 (1903), 554–558.Google Scholar
11. Mittag-Leffler, G. M., Sur la représentation analytique d'une branche uniforme d'une fonction monogène, Acta Math. 29 (1905), 101–181.Google Scholar
12. Oldham, K. B. and Spanier, J., The fractional calculus (Academic Press, 1974).Google Scholar
13. Ostrovskii, I. V. and Peresyolkova, I. N., Nonasymptotic results on distribution of zeros of the function E ρ(z, μ), Analysis Math. 23 (1997), 283–296.Google Scholar
14. Podlubny, I., Fractional differential equations (Academic Press, 1999).Google Scholar
15. Schneider, W. R., Fractional diffusion, in Dynamics and stochastic processes: theory and applications, Lecture Notes in Physics, Volume 355, pp. 276–286 (Springer, 1990).Google Scholar
16. Schneider, W. R. and Wyss, W., Fractional diffusion and wave equations, J. Math. Phys. 30 (1989), 134–144.Google Scholar
17. Vabishchevich, P. N., Nonlocal parabolic problems and the inverse heat-conduction problem, Diff. Uravn. 17 (1981), 1193–1199.Google Scholar
18. Wang, R. N., Chen, D. H. and Xiao, T. J., Abstract fractional Cauchy problems with almost sectorial operators, J. Diff. Eqns 252 (2012), 202–235.Google Scholar
19. Wang, J., Wei, W. and Fečkan, M., Nonlocal Cauchy problems for fractional evolution equations involving Volterra–Fredholm type integral operators, Miskolc Math. Notes 13(1) (2012), 127–147.Google Scholar
20. Zhou, Y. and Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlin. Analysis 11 (2010), 4465–4475.Google Scholar