DOI: 10.1515/anona-2023-0121 ISSN: 2191-950X

Blowup in L 1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms

Giuseppe Floridia, Yikan Liu, Masahiro Yamamoto
  • Analysis


This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity

u p {u}^{p}
in a bounded domain
Ω \Omega
with the homogeneous Neumann boundary condition and positive initial values. In the case of
p > 1 p\gt 1
, we prove the blowup of solutions
u ( x , t ) u\left(x,t)
in the sense that
u ( , t ) L 1 ( Ω ) \Vert u\left(\hspace{0.33em}\cdot \hspace{0.33em},t){\Vert }_{{L}^{1}\left(\Omega )}
tends to
t t
approaches some value, by using a comparison principle for the corresponding ordinary differential equations and constructing special lower solutions. Moreover, we provide an upper bound for the blowup time. In the case of
0 < p < 1 0\lt p\lt 1
, we establish the global existence of solutions in time based on the Schauder fixed-point theorem.

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