The Critical Exponent(s) for the Semilinear Fractional Diffusive Equation

Abstract

In this paper we show that there exist two different critical exponents for global small data solutions to the semilinear fractional diffusive equationwhere α ∈ (0, 1), and ∂ 1+α t u is the Caputo fractional derivative in time. The second critical exponent appears if the second data is assumed to be zero. This peculiarity is related to the fact that the order of the equation is fractional, and so the role played by the second data u 1 becomes "unnatural" as α decreases to zero. To prove our result, we first derive L r − L q linear estimates, 1 ≤ r ≤ q ≤ ∞, for the solution to the linear Cauchy problem, where |u| p is replaced by f (t, x), and then we apply a contraction argument.2010 Mathematics Subject Classification. Primary 35R11; Secondary 35A01, 35B33.

Global small data solutions for semilinear waves with two dissipative terms

“…For this reason, to deal with the critical case, we do not rely on estimate of |u(t, ⋅)| p in L , but in some L 1 spaces with 1 ∈ [1, ) . The same strategy is used in the context of semilinear damped waves in [30] and in the context of semilinear fractional diffusive equations in [31].…”

Semi-linear fractional $$\varvec{\sigma }$$-evolution equations with mass or power non-linearity

“…The following result was proposed and proved by Marcello D'Abbicco (University of Bari) and already used in a special case in [4]. We present the proof to make the paper more self-contained.…”

Global existence for semi-linear time fractional σ-evolution equations

“…Specifically, in the 1980s, Glassey 5 proposed the conjecture that 𝑝 𝐺𝑙𝑎 (𝑛) = 1 + 2 𝑛 − 1 is the critical exponent for the Cauchy problem (4), it means that in the supercritical case 𝑝 > 𝑝 𝐺𝑙𝑎 , small data solutions exist globally, meanwhile the global solutions dose not exist under proper assumptions about initial data in the case of 𝑝 ≤ 𝑝 𝐺𝑙𝑎 . Kunio 6 and Tzvetkov 7 researched global existence and asymptotic behavior of solutions for general data as 𝑝 > 𝑝 𝐺𝑙𝑎 , in the case of space dimension 𝑛 = 2, 3, respectively.…”

“…Recently, D'Abbicco et al 4 studied the Cauchy problem for the following semi-linear time fractional diffusion-wave equation…”

“…1,2,4,5 here we attempt to study semi-linear time fractional 𝜎−evolution equation models with new nonlinearities |𝑢 𝑡 | 𝑝 and ∫ 𝜏) −𝜇 |𝑢 𝑡 (𝜏, 𝑥)| 𝑝 𝑑𝜏. Note that the second power nonlinearity is the Riemann-Liouville fractional integral of |𝑢 𝑡 | 𝑝 , and we remark that…”