Local existence for evolution equations with nonlocal term in time and singular initial data

Aldryn Aparcana; Ricardo Castillo; Omar Guzmán-Rea; Miguel Loayza

doi:10.1007/s00033-022-01723-x

Local existence for evolution equations with nonlocal term in time and singular initial data

Aldryn Aparcana, Ricardo Castillo, Omar Guzmán-Rea, Miguel Loayza

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

Abstract

We consider the semilinear equation ut+(-Δ)α/2u=∫0tm(t,s)f(u(s))dsin Ω × (0 , T) , where 0 < α≤ 2 , m is a nonnegative and measurable homogeneous function defined on K= { (t, s) ∈ R², 0 < s< t} , f is a nonnegative, continuous and nondecreasing function and Ω is either a bounded smooth domain or the whole space R^N. Our goal is to determine conditions for the local existence and nonexistence of solutions with nonnegative initial data belonging to the space L^r(Ω) , 1 ≤ r< ∞.

Original language	English
Article number	85
Journal	Zeitschrift fur Angewandte Mathematik und Physik
Volume	73
Issue number	2
DOIs	https://doi.org/10.1007/s00033-022-01723-x
State	Published - Apr 2022

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Keywords

Fractional heat equation
Local existence
Nonexistence
Nonlocal parabolic equation
Singular initial data

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10.1007/s00033-022-01723-x

Cite this

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abstract = "We consider the semilinear equation ut+(-Δ)α/2u=∫0tm(t,s)f(u(s))dsin Ω × (0 , T) , where 0 < α≤ 2 , m is a nonnegative and measurable homogeneous function defined on K= { (t, s) ∈ R2, 0 < s< t} , f is a nonnegative, continuous and nondecreasing function and Ω is either a bounded smooth domain or the whole space RN. Our goal is to determine conditions for the local existence and nonexistence of solutions with nonnegative initial data belonging to the space Lr(Ω) , 1 ≤ r< ∞.",

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AU - Aparcana, Aldryn

AU - Castillo, Ricardo

AU - Guzmán-Rea, Omar

AU - Loayza, Miguel

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N2 - We consider the semilinear equation ut+(-Δ)α/2u=∫0tm(t,s)f(u(s))dsin Ω × (0 , T) , where 0 < α≤ 2 , m is a nonnegative and measurable homogeneous function defined on K= { (t, s) ∈ R2, 0 < s< t} , f is a nonnegative, continuous and nondecreasing function and Ω is either a bounded smooth domain or the whole space RN. Our goal is to determine conditions for the local existence and nonexistence of solutions with nonnegative initial data belonging to the space Lr(Ω) , 1 ≤ r< ∞.

AB - We consider the semilinear equation ut+(-Δ)α/2u=∫0tm(t,s)f(u(s))dsin Ω × (0 , T) , where 0 < α≤ 2 , m is a nonnegative and measurable homogeneous function defined on K= { (t, s) ∈ R2, 0 < s< t} , f is a nonnegative, continuous and nondecreasing function and Ω is either a bounded smooth domain or the whole space RN. Our goal is to determine conditions for the local existence and nonexistence of solutions with nonnegative initial data belonging to the space Lr(Ω) , 1 ≤ r< ∞.

KW - Fractional heat equation

KW - Local existence

KW - Nonexistence

KW - Nonlocal parabolic equation

KW - Singular initial data

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Local existence for evolution equations with nonlocal term in time and singular initial data

Abstract

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Keywords

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