No local L1 solutions for semilinear fractional heat equations

Kexue Li

doi:10.1515/fca-2017-0070

No local L¹ solutions for semilinear fractional heat equations

Kexue Li

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We study the Cauchy problem for the semilinear fractional heat equation u_t = Δ^α/2 u + f(u) with non-negative initial value u₀ ϵ L^q(ℝⁿ) and locally Lipschitz, non-decreasing, non-negative source term f. For f satisfying the Osgood-type condition f1∞dsf(s)=∞, we show that there exist initial conditions such that the equation has no local solution in Lloc1(Rn).

Original language	English
Pages (from-to)	1328-1337
Number of pages	10
Journal	Fractional Calculus and Applied Analysis
Volume	20
Issue number	6
DOIs	https://doi.org/10.1515/fca-2017-0070
State	Published - 20 Dec 2017

Keywords

fractional heat equation
fractional heat kernel
local existence
non-existence of solutions

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title = "No local L1 solutions for semilinear fractional heat equations",

abstract = "We study the Cauchy problem for the semilinear fractional heat equation ut = Δα/2 u + f(u) with non-negative initial value u0 ϵ Lq(ℝn) and locally Lipschitz, non-decreasing, non-negative source term f. For f satisfying the Osgood-type condition f1∞dsf(s)=∞, we show that there exist initial conditions such that the equation has no local solution in Lloc1(Rn).",

keywords = "fractional heat equation, fractional heat kernel, local existence, non-existence of solutions",

author = "Kexue Li",

note = "Publisher Copyright: {\textcopyright} 2017 Diogenes Co., Sofia 2017.",

year = "2017",

month = dec,

day = "20",

doi = "10.1515/fca-2017-0070",

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AU - Li, Kexue

PY - 2017/12/20

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N2 - We study the Cauchy problem for the semilinear fractional heat equation ut = Δα/2 u + f(u) with non-negative initial value u0 ϵ Lq(ℝn) and locally Lipschitz, non-decreasing, non-negative source term f. For f satisfying the Osgood-type condition f1∞dsf(s)=∞, we show that there exist initial conditions such that the equation has no local solution in Lloc1(Rn).

AB - We study the Cauchy problem for the semilinear fractional heat equation ut = Δα/2 u + f(u) with non-negative initial value u0 ϵ Lq(ℝn) and locally Lipschitz, non-decreasing, non-negative source term f. For f satisfying the Osgood-type condition f1∞dsf(s)=∞, we show that there exist initial conditions such that the equation has no local solution in Lloc1(Rn).

KW - fractional heat equation

KW - fractional heat kernel

KW - local existence

KW - non-existence of solutions

UR - https://www.scopus.com/pages/publications/85041959199

U2 - 10.1515/fca-2017-0070

DO - 10.1515/fca-2017-0070

M3 - 文章

AN - SCOPUS:85041959199

SN - 1311-0454

VL - 20

SP - 1328

EP - 1337

JO - Fractional Calculus and Applied Analysis

JF - Fractional Calculus and Applied Analysis

IS - 6

ER -

No local L¹ solutions for semilinear fractional heat equations

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