Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms

Abstract

This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity up{u}^{p} in a bounded domain Ω\Omega with the homogeneous Neumann boundary condition and positive initial values. In the case of p>1p\gt 1, we prove the blowup of solutions u(x,t)u\left(x,t) in the sense that ‖u(⋅,t)‖L1(Ω)\Vert u\left(\hspace{0.33em}\cdot \hspace{0.33em},t){\Vert }_{{L}^{1}\left(\Omega )} tends to ∞\infty as tt 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<10\lt p\lt 1, we establish the global existence of solutions in time based on the Schauder fixed-point theorem.

Keywords

• semilinear time-fractional diffusion equation
• polynomial nonlinearity
• blowup
• global existence
• 35r11
• 35k58
• 35b44