Advances in Differential Equations

Quenching rate of solutions for a semilinear parabolic equation

Masaki Hoshino

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We study the behavior of solutions of the Cauchy problem for a semilinear parabolic equation with a singular absorption term. We discuss the convergence of solutions to a singular stationary solution from above as time goes to infinity, and show that in a supercritical case a sharp estimate of the quenching rate can be determined explicitly when a specific growth rate of initial data is given. We also obtain a universal lower bound of the quenching rate which implies the optimality of the results. Proofs are given by a comparison method that is based on matched asymptotic expansion. We first determine a quenching rate of solutions by a formal analysis. Based on the formal analysis, we give a rigorous proof by constructing appropriate super- and subsolutions with the desired quenching rate.

Article information

Adv. Differential Equations, Volume 16, Number 5/6 (2011), 401-434.

First available in Project Euclid: 17 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K15: Initial value problems for second-order parabolic equations 35B35: Stability 35B40: Asymptotic behavior of solutions


Hoshino, Masaki. Quenching rate of solutions for a semilinear parabolic equation. Adv. Differential Equations 16 (2011), no. 5/6, 401--434.

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