Stable blow-up patterns

Alberto Bressan

doi:10.1016/0022-0396(92)90104-U

Stable blow-up patterns

Alberto Bressan

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

45 Scopus citations

Abstract

For the semilinear heat equation u_t = Δu + e^u in a convex domain Ω ⊂ Rⁿ, given any b ε{lunate} Ω we show the existence of solutions which blow up in finite time exactly at b and whose final profile has the form u(T, x) ≈ -2 ln |x - b| + ln |ln |x - b|| + ln 8, T being the blow-up time. Using a suitable set of rescaled coordinates, this asymptotic behavior is proved to be stable with respect to small perturbations of the initial conditions.

Original language	English (US)
Pages (from-to)	57-75
Number of pages	19
Journal	Journal of Differential Equations
Volume	98
Issue number	1
DOIs	https://doi.org/10.1016/0022-0396(92)90104-U
State	Published - Jul 1992

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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10.1016/0022-0396(92)90104-U

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title = "Stable blow-up patterns",

abstract = "For the semilinear heat equation ut = Δu + eu in a convex domain Ω ⊂ Rn, given any b ε{lunate} Ω we show the existence of solutions which blow up in finite time exactly at b and whose final profile has the form u(T, x) ≈ -2 ln |x - b| + ln |ln |x - b|| + ln 8, T being the blow-up time. Using a suitable set of rescaled coordinates, this asymptotic behavior is proved to be stable with respect to small perturbations of the initial conditions.",

author = "Alberto Bressan",

year = "1992",

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AB - For the semilinear heat equation ut = Δu + eu in a convex domain Ω ⊂ Rn, given any b ε{lunate} Ω we show the existence of solutions which blow up in finite time exactly at b and whose final profile has the form u(T, x) ≈ -2 ln |x - b| + ln |ln |x - b|| + ln 8, T being the blow-up time. Using a suitable set of rescaled coordinates, this asymptotic behavior is proved to be stable with respect to small perturbations of the initial conditions.

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JO - Journal of Differential Equations

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ER -

Stable blow-up patterns

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