Advances in Differential Equations

Local and global properties of solutions of heat equation with superlinear absorption

Tai Nguyen Phuoc and Laurent Véron

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Abstract

We study the limit when $k\to\infty$ of the solutions of $ \partial_tu-\Delta u+f(u)=0$ in $\mathbb R^N\times (0,\infty)$ with initial data $k\delta$, when $f$ is a positive superlinear increasing function. We prove that there exist essentially three types of possible behaviour according to whether $f^{-1}$ and $F^{-1/2}$ belong or not to $L^1(1,\infty)$, where $F(t)=\int_0^t f(s)ds$. We use these results for providing a new and more general construction of the initial trace and some uniqueness and nonuniqueness results for solutions with unbounded initial data.

Article information

Source
Adv. Differential Equations Volume 16, Number 5/6 (2011), 487-522.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355703298

Mathematical Reviews number (MathSciNet)
MR2816114

Zentralblatt MATH identifier
1222.35109

Subjects
Primary: 35K58: Semilinear parabolic equations 35K91: Semilinear parabolic equations with Laplacian, bi-Laplacian or poly- Laplacian 35K15: Initial value problems for second-order parabolic equations

Citation

Phuoc, Tai Nguyen; Véron, Laurent. Local and global properties of solutions of heat equation with superlinear absorption. Adv. Differential Equations 16 (2011), no. 5/6, 487--522. https://projecteuclid.org/euclid.ade/1355703298.


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