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

Giuseppe Floridia, Yikan Liu, Masahiro Yamamoto

TL;DR

The paper studies semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain under homogeneous Neumann conditions, for $0<\alpha<1$. It shows blowup in the $L^1(\Omega)$-norm when $p>1$ and establishes global existence for $0<p<1$, using a reduction to a time-fractional ODE via the integral against the first eigenfunction and constructing a lower solution for the blowup case, as well as a Schauder-fixed point approach for the sublinear case. A sharp upper bound on the blowup time, $T^*(\alpha,p,a)$, is derived and shown to generalize the parabolic limit as $\alpha\uparrow1$, while the global existence result leverages mild solution representations with Mittag-Leffler kernels $E_{\alpha,\beta}$ and a compactness argument for the nonlinear term. The results provide a fractional-extension of classical blowup theory and offer a methodological framework applicable to other nonlinearities and boundary conditions, with potential extensions to more general operators and nonlinearities.

Abstract

This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $Ω$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$, we prove the blowup of solutions $u(x,t)$ in the sense that $\|u(\,\cdot\,,t)\|_{L^1(Ω)}$ tends to $\infty$ as $t$ 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 $0<p<1$, we establish the global existence of solutions in time based on the Schauder fixed-point theorem.

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

TL;DR

The paper studies semilinear time-fractional diffusion equations with polynomial nonlinearity in a bounded domain under homogeneous Neumann conditions, for . It shows blowup in the -norm when and establishes global existence for , using a reduction to a time-fractional ODE via the integral against the first eigenfunction and constructing a lower solution for the blowup case, as well as a Schauder-fixed point approach for the sublinear case. A sharp upper bound on the blowup time, , is derived and shown to generalize the parabolic limit as , while the global existence result leverages mild solution representations with Mittag-Leffler kernels and a compactness argument for the nonlinear term. The results provide a fractional-extension of classical blowup theory and offer a methodological framework applicable to other nonlinearities and boundary conditions, with potential extensions to more general operators and nonlinearities.

Abstract

This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity in a bounded domain with the homogeneous Neumann boundary condition and positive initial values. In the case of , we prove the blowup of solutions in the sense that tends to as 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 , we establish the global existence of solutions in time based on the Schauder fixed-point theorem.
Paper Structure (5 sections, 6 theorems, 102 equations)

This paper contains 5 sections, 6 theorems, 102 equations.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries
  3. Completion of proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['thm2']}
  5. Concluding remarks and discussions

Key Result

Theorem 1

Let $p>1$ and $a\in H^{2\gamma}(\Omega)$ satisfy $\partial_\nu a=0$ on $\partial\Omega$ and $a\ge0,\not\equiv0$ in $\Omega$. Then, there exists some $T=T_{\alpha,p,a} >0$ such that the solution satisfying 1.2 exists for $0<t<T_{\alpha,p,a}$ and eq-blowup holds. Moreover, we can bound $T_{\alpha,p,a}

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 2
  • Lemma 3
  • proof
  • ...and 3 more