Abstract
We consider the following fractional reaction-diffusion equation
gα(t) = tα−1/Γ(α) (0 < α < 1), f ∈ C([0,∞)) is an elliptic operator whose fundamental solution of its associated parabolic equation has Gaussian lower and upper bounds. We characterize the behavior of the functions f so that the above fractional reaction-diffusion equation has a bounded local solution in Lr(Ω), for non-negative initial data u0 ∈ Lr(Ω), when r > 1 and Ω ⊂ ℝN is either a smooth bounded domain or the whole space ℝN. The case r = 1 is also studied.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Aparcana, R. Castillo, O. Guzmán-Rea, M. Loayza, On the local existence for a weakly parabolic system in Lebesgue spaces. J. Differential Equations 268, No 6 (2020), 3129–3151.
W. Arendt, A.F.M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions. J. Operator Theory 38, No 1 (1997), 87–130.
D.G. Aronson, Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc 73,(1967), 890–896.
D.G. Aronson, Non-negative solutions of linear parabolic equations. Ann. ScuolaNorm. Sup. Pisa Cl. Sci. (3) 22,(1968), 607–694.
E.G. Bazhlekova, Subordination principle for fractional evolution equations. Fract. Calc. Appl. Anal 3, No 3 (2000), 213–230.
H. Brezis, T. Cazenave, A nonlinear heat equation with singular initial data. J. Anal. Math 68,(1996), 277–304.
C. Celik, Z. Zhou, No local L1 solution for a nonlinear heat equation. Commun. Partial Differ. Equ 28,(2003), 1807–1831.
M. Choulli, L. Kayser, Gaussian lower bound for the Neumann Green function of a general parabolic operator. Positivity 19, No 3 (2015), 625–646.
B. de Andrade, A. Viana, On a fractional reaction-diffusion equation. Z. Angew. Math. Phys 68, No 3 (2017), Art. 59, 11.
B. de Andrade, G. Siracusa, A. Viana, A nonlinear fractional diffusion equation: well-posedness, comparison results and blow-up (2020), Submitted.
Y. Fujishima, N. Ioku, Existence and non-existence of solutions for the heat equation with a superlinear source term. J. Math. Pures Appl. (9) 118,(2018), 128–158.
H. Fujita, S. Watanabe, On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations. Comm. Pure Appl. Math 21,(1968), 631–652.
J. Kemppainen, J. Siljander, V. Vergara, R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in ℝd. Math. Ann 366, No 3-4 (2016), 941–979.
R. Laister, J.C. Robinson, M. Sierzega, A. Vidal-López, A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces. Ann. Inst. H. Poincaré Anal.Non Linéaire 33,(2016), 1519–1538.
R. Laister, J.C. Robinson, M. Sierzega, A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations. J. Differential Equations 262, No 10 (2017), 4979–4987.
K. Li, A characteristic of local existence for nonlinear fractional heat equations in Lebesgue spaces. Comput. Math. Appl 73, No 4 (2017), 653–665.
A. Lopushansky, O. Lopushansky, A. Szpila, Fractional abstract Cauchy problem on complex interpolation scales. Fract. Calc. Appl. Anal 23, No 4 (2020), 1125–1140; DOI: 10.1515/fca-2020-0057 https://www.degruyter.com/journal/key/FCA/23/4/html
R. Metzler, E. Barkai, J. Klafter, Anomalous transport in disordered systems underthe influence of external fields. Physica A 266,(1999), 343–350.
R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37, No 31 (2004), R161–R208.
J. Peng, K. Li, A novel characteristic of solution operator for the fractional abstract Cauchy problem. J. Math. Anal. Appl 385,(2012), 786–796.
J.C. Robinson, M. Sierzega, Supersolutions for a class of semilinear heat equations. Rev. Mat. Complut 26,(2013), 341–360.
W.R. Schneider, W. Wyss, Fractional diffusion and wave equations. J. Math. Phys 30, No 1 (1989), 134–144.
A. Viana, A local theory for a fractional reaction-diffusion equation. Commun. Contemp. Math 21, No 6 (2019), # 1850033, 26.
F.B. Weissler, Semilinear evolution equations in Banach spaces. J. Funct. Anal 32,(1979), 277–296.
F.B. Weissler, Local existence and nonexistence for semilinear parabolic eequations in Lp. Indiana Univ. Math. J 29, No 1 (1980), 79–102.
F.B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation. Isr. J. Math 38,(1981), 29–40.
Q.-G. Zhang, H.-R. Sun, The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation. Topol. MethodsNonlinear Anal 46, No 1 (2015), 69–92.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Castillo, R., Loayza, M. & Viana, A. Local Existence and Non-Existence for a Fractional Reaction-Diffusion Equation in Lebesgue Spaces. Fract Calc Appl Anal 24, 1193–1219 (2021). https://doi.org/10.1515/fca-2021-0051
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1515/fca-2021-0051