Applicable Analysis Vol. 91, No. 12, December 2012, 2267–2276 Semilinear heat equation with absorption and a nonlocal boundary condition Alexander Gladkov a * and Mohammed Guedda b a Department of Mathematics, Vitebsk State University, Moskovskii pr. 33, Vitebsk 210038, Belarus; b LAMFA, CNRS, UMR 6140, Universite´ de Picardie, 33 rue Saint-Leu, Amiens, F-80039, France Communicated by A. Pankov (Received 31 December 2010; final version received 19 June 2011) We consider a semilinear parabolic equation u t ¼ Du  c(x, t)u p for (x, t) 2   (0, T ) with nonlinear and nonlocal boundary condition uj @  [0,T ] ¼ R  k(x, y, t)u l dy and nonnegative initial data where p 4 0 and l 4 0. We first establish local existence theorem and comparison principle. Then we prove uniqueness of solutions with trivial initial datum for ( p þ 1)/ 2 5 l 5 1, with nonnegative initial data for l  1, with positive initial data under the conditions l 5 1, p  1 as well as positive in Q T solutions if max( p, l) 5 1, nonuniqueness of solution with trivial initial datum for l 5 min{1, ( p þ 1)/2}. Keywords: semilinear parabolic equation; nonlocal boundary condition; uniqueness AMS Subject Classifications: 35K20; 35K58; 35K61 1. Introduction In this article we consider the following nonlocal initial boundary value problem: u t ¼ Du  cðx, tÞu p , for x 2 ,0 5 t 5 T, uðx, tÞ¼ Z  kðx, y, tÞu l ð y, tÞdy, for x 2 @,0 5 t 5 T, uðx,0Þ¼ u 0 ðxÞ, for x 2 , 8 > > < > > : ð1:1Þ where  is a bounded domain in R n for n  1 with smooth boundary @, T 4 0, p 4 0 and l 4 0. Here, c(x, t) is a nonnegative Ho¨ lder continuous function defined for x 2  and t 2 [0, T ] and k(x, y, t) is a nonnegative continuous function defined for x 2 @, y 2  and t 2 [0, T ]. The initial data u 0 (x) is a nonnegative continuous function satisfying the boundary condition at t ¼ 0. Over the last 30 years, many physical phenomena were formulated into nonlocal mathematical models [1–6]. Initial boundary value problem for diffusion and *Corresponding author. Email: gladkoval@mail.ru ISSN 0003–6811 print/ISSN 1563–504X online ß 2012 Taylor & Francis http://dx.doi.org/10.1080/00036811.2011.601297 http://www.tandfonline.com