The critical exponent for a time fractional diffusion equation with nonlinear memory

Abstract

In this paper, we determine the Fujita critical exponent of the following time fractional subdiffusion equation with nonlinear memoryany nontrivial positive solution blows up in a finite time. If p > p * and ||u 0 || L q c (R N ) is sufficiently small, where q c = N (p−1) 2( + ) , then u exists globally. KEYWORDS blow-up, Fujita critical exponent, global existence, nonlinear memory, time fractional diffusion equation Math Meth Appl Sci. 2018;41:6443-6456. wileyonlinelibrary.com/journal/mma

On the critical exponents for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain

“…Proof. The proofs of Property (i) and (ii) are similar to those of Lemma 5(i),(ii) and (iv) in [32]. For the convenience of the reader and the completeness of the paper, here we give sketchy proofs of Property (i) and (ii).…”

“…In order to prove blow-up results by the eigenfunction method due to [14], we prove the following results on an ordinary fractional differential equation, which are similar to ones in [5,32].…”

“…Motivated by the above results, in this paper, we investigate sharp blow-up and global existence of solutions for problem (1.1), and then extend the results in [5,32,33] to the time fractional case (1 < α < 2). Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term(i.e.…”

“…In recent years, fractional differential equations have gained considerable popularity and importance, due to their demonstrated applications in seemingly widespread fields of science and engineering [15,21]. Hence, recently, there are a lot of papers on the existence and properties of solutions for fractional differential equations [1][2][3][4]12,16,19,[25][26][27][28][29][31][32][33][34][35][36].…”

“…In [5], Cazenave et al studied the following semilinear heat equation with a nonlinear memory term u t − △u = 0 I 1−γ t (|u| p−1 u) on both R N and a bounded domain Ω ⊂ R N , and obtained the critical exponents of this problem. Recently, the authors of [20,32,33] generalized the results of [5] to the time fractional case (0 < α < 1), and gave the critical exponents of this problem when α < γ and α ≥ γ, respectively. In [6], Chen and Palmieri established a generalized Strauss exponent p 0 (N, γ) for problem (1.1) with α = 2 on R N .…”