Semilinear heat equation with absorption and a nonlocal boundary condition

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}.

Keywords:

• semilinear parabolic equation
• nonlocal boundary condition
• uniqueness

AMS Subject Classifications::

• 35K20
• 35K58
• 35K61

Acknowledgement

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