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Applicable Analysis
An International Journal
Volume 91, 2012 - Issue 12
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Original Articles

Semilinear heat equation with absorption and a nonlocal boundary condition

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Pages 2267-2276 | Received 31 Dec 2010, Accepted 19 Jun 2011, Published online: 19 Sep 2011
 

Abstract

We consider a semilinear parabolic equation u t  = Δu − c(x, t)u p for (x, t) ∈ Ω × (0, T) with nonlinear and nonlocal boundary condition u|∂Ω × [0,T] = ∫Ω k(x, y, t)u l  dy and nonnegative initial data where p > 0 and l > 0. We first establish local existence theorem and comparison principle. Then we prove uniqueness of solutions with trivial initial datum for (p + 1)/2 < l < 1, with nonnegative initial data for l ≥ 1, with positive initial data under the conditions l < 1, p ≥ 1 as well as positive in solutions if max(p, l) < 1, nonuniqueness of solution with trivial initial datum for l < min{1, (p + 1)/2}.

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Acknowledgement

This work was supported in part by DAI, UPJV Amiens and the Regional Council of Picardie.

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