On the behavior of solutions for large |x| of parabolic equations with unbounded coefficients
Lu-San Chen · Tohoku Mathematical Journal · 1968
Let R n be the w-dimensional Euclidean space whose points x are represented by its coordinates (x ί9 , x n ) and let Ω Γ = R n x(0, T) (T) be a strip in the (n + l)-dimensional Euclidean half-space i? w x(0, oo).Every point in Ω Γ is denoted by (x, t), xzR n , te (0, T).We introduce a function space E\(β τ )(\ £ (0, 1]) which is the totality of functions W(x, t) such that in the closure Q T of Ω τ for some positive constants μ and cc.Consider a parabolic differential equation /IN T NΓ-3 2 w , ^ , 3w , du (1) i ΛS Σ βfJ ^_ § _ + g ftt __.+ αt with variable coefficients a tj (= α u ), έ 4 and c defined in £} τ , where n Σ2 a i5%ι%i > 0 in χ5 Γ for every non-zero real vector £=(£i,•••,£,»)• We assume that there exist positive constants JK^, JK 2> K Z and λ ^ (0,1] such that in Q τ (2) Σ i,i (3) |έ t (4)Under these assumptions the equation ( 1) was treated by many authors, Krzyzaήski, Bodanko, Aronson, Besala and others.In particular, Bodanko [2]