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References (20)
-
J. Takahashi (2015)
Solvability of a Semilinear Parabolic Equation with Measures as Initial Data -
(2003)
No local L1 solution for a nonlinear heat equation, Comm -
H. Fujita (1966)
On the blowing up of solutions fo the Cauchy problem for u_t=Δu+u^, 13
-
C. Çelik, Zhengfang Zhou (2003)
No Local L 1Solution for a Nonlinear Heat EquationCommunications in Partial Differential Equations, 28
-
F. Weissler (1981)
Existence and non-existence of global solutions for a semilinear heat equationIsrael Journal of Mathematics, 38
-
R. Laister, James Robinson, M. Sierżęga, Alejandro Vidal-L'opez (2014)
A complete characterisation of local existence for semilinear heat equations in Lebesgue spacesAnnales de l'Institut Henri Poincaré C, Analyse non linéaire, 33
-
(2007)
Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts: Basler Lehrbücher -
(1980)
Local existence and nonexistence for semilinear parabolic equations in L -
P. Quittner, P. Souplet (2007)
Superlinear Parabolic ProblemsBirkhäuser Advanced Texts Basler Lehrbücher
-
Kotaro Hisa, Kazuhiro Ishige (2016)
Existence of solutions for a fractional semilinear parabolic equation with singular initial dataNonlinear Analysis
-
(1981)
Existence and nonexistence of global solutions for a semilinear heat equation, Israel -
(1981)
Existence and nonexistence of global solutions for a semilinear heat equation -
(2002)
Local existence and nonexistence for semilinear parabolic equations in L p -
W. Ni, P. Sacks (1985)
Singular behavior in nonlinear parabolic equationsTransactions of the American Mathematical Society, 287
-
H. Brezis, T. Cazenave (1996)
A nonlinear heat equation with singular initial dataJournal d’Analyse Mathématique, 68
-
P. Baras, M. Pierre (1985)
Critère d'existence de solutions positives pour des équations semi-linéaires non monotonesAnnales De L Institut Henri Poincare-analyse Non Lineaire, 2
-
James Robinson, M. Sierżęga (2011)
Supersolutions for a class of semilinear heat equationsRevista Matemática Complutense, 26
-
E. Terraneo (2002)
NON-UNIQUENESS FOR A CRITICAL NON-LINEAR HEAT EQUATIONCommunications in Partial Differential Equations, 27
-
(2016)
Vidal-López, A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces, Ann -
(1982)
Nonuniqueness for a semilinear initial value problem
- Publisher
- Springer Journals
- Copyright
- Copyright © Springer Nature Switzerland AG 2020
- ISSN
- 1424-3199
- eISSN
- 1424-3202
- DOI
- 10.1007/s00028-020-00573-2
- Publisher site
- See Article on Publisher Site
Abstract
We study the existence and nonexistence for a Cauchy problem of the semilinear heat equation: ∂tu=Δu+|u|p-1uinRN×(0,T),u(x,0)=ϕ(x)inRN\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu=\Delta u+|u|^{p-1}u &{} \text {in}\ \mathbb {R}^N\times (0,T),\\ u(x,0)=\phi (x) &{} \text {in}\ \mathbb {R}^N\end{array}\right. } \end{aligned}$$\end{document}in L1(RN)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^1(\mathbb {R}^N)$$\end{document}. Here, N≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N\ge 1$$\end{document}, p=1+2/N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p=1+2/N$$\end{document} and ϕ∈L1(RN)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\phi \in L^1(\mathbb {R}^N)$$\end{document} is a possibly sign-changing initial function. Since N(p-1)/2=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N(p-1)/2=1$$\end{document}, the L1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^1$$\end{document} space is scale critical and this problem is known as a doubly critical case. It is known that a solution does not necessarily exist for every ϕ∈L1(RN)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\phi \in L^1(\mathbb {R}^N)$$\end{document}. Let Xq:={ϕ∈Lloc1(RN)|∫RN|ϕ|log(e+|ϕ|)qdx<∞}(⊂L1(RN))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_q:=\{\phi \in L^1_\mathrm{{loc}}(\mathbb {R}^N)\ |\ \int _{\mathbb {R}^N}|\phi |\left[ \log (e+|\phi |)\right] ^q\mathrm{d}x<\infty \}(\subset L^1(\mathbb {R}^N))$$\end{document}. In this paper, we construct a local-in-time mild solution in L1(RN)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^1(\mathbb {R}^N)$$\end{document} for ϕ∈Xq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\phi \in X_q$$\end{document} if q≥N/2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q\ge N/2$$\end{document}. We show that, for each 0≤q<N/2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0\le q<N/2$$\end{document}, there is a nonnegative initial function ϕ0∈Xq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\phi _0\in X_q$$\end{document} such that the problem has no nonnegative solution, using a necessary condition given by Baras–Pierre (Ann Inst Henri Poincaré Anal Non Linéaire 2:185–212, 1985). Since Xq⊂XN/2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_q\subset X_{N/2}$$\end{document} for q≥N/2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q\ge N/2$$\end{document}, XN/2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_{N/2}$$\end{document} becomes a sharp integrability condition. We also prove a uniqueness in a certain set of functions which guarantees the uniqueness of the solution constructed by our method.
Journal
Journal of Evolution Equations – Springer Journals
Published: Apr 1, 2020