Sign-changing solutions of the nonlinear heat equation with persistent singularities
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 126

We study the existence of sign-changing solutions to the nonlinear heat equation ∂$$u = Δu + |u|$$u on ℝ$$, N ≥ 3, with $$, where $$, which are singular at x = 0 on an interval of time. In particular, for certain μ > 0 that can be arbitrarily large, we prove that for any $$ which is bounded at infinity and equals $$ in a neighborhood of 0, there exists a local (in time) solution u of the nonlinear heat equation with initial value u0, which is sign-changing, bounded at infinity and has the singularity $$ at the origin in the sense that for t > 0, $$ as |x|→ 0, where $$. These solutions in general are neither stationary nor self-similar.

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DOI : 10.1051/cocv/2020082
Classification : 35K91, 35K58, 35C06, 35K67, 35A01, 35A21
Keywords: Nonlinear heat equation, sign-changing solutions, singular self-similar solutions, singular stationary solutions, persistent singularities 
@article{COCV_2020__26_1_A126_0,
     author = {Cazenave, Thierry and Dickstein, Fl\'avio and Naumkin, Ivan and Weissler, Fred B.},
     editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinsk, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
     title = {Sign-changing solutions of the nonlinear heat equation with persistent singularities},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     year = {2020},
     publisher = {EDP Sciences},
     volume = {26},
     doi = {10.1051/cocv/2020082},
     mrnumber = {4188829},
     zbl = {1459.35249},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv/2020082/}
}
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TI  - Sign-changing solutions of the nonlinear heat equation with persistent singularities
JO  - ESAIM: Control, Optimisation and Calculus of Variations
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%E Casas, E.
%E de Teresa, L.
%E Glowinsk, R.
%E Leugering, G.
%E Trélat, E.
%E Zhang, X.
%T Sign-changing solutions of the nonlinear heat equation with persistent singularities
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2020
%V 26
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%R 10.1051/cocv/2020082
%G en
%F COCV_2020__26_1_A126_0
Cazenave, Thierry; Dickstein, Flávio; Naumkin, Ivan; Weissler, Fred B. Sign-changing solutions of the nonlinear heat equation with persistent singularities. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 126. doi: 10.1051/cocv/2020082

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Research supported by the “Brazilian-French Network in Mathematics”.

Flavio Dickstein was partially supported by CNPq (Brasil).

Ivan Naumkin is a Fellow of Sistema Nacional de Investigadores. He was partially supported by project PAPIIT IA101820.