DOI 10.1007/s10958-018-3876-z Journal of Mathematical Sciences, Vol. 232, No. 3, July, 2018 THE STABILIZATION RATE OF SOLUTIONS TO THE CAUCHY PROBLEM FOR PARABOLIC EQUATIONS V. N. Denisov Lomonosov Moscow State University Moscow 119991, Russia firstname.lastname@example.org UDC 517.956.4 We study suﬃcient conditions on lower-order coeﬃcients of a nondivergence-form parabolic equation that guarantee the power rate of the uniform stabilization of the so- lution to the Cauchy problem on every compact set K of R for any bounded initial function. Bibliography:7 titles. In memory of V. V. Zhikov 1 Introduction In the half-space D = R × (0, ∞), N 3, we consider the Cauchy problem L u ≡ Lu +(b, ∇u)+ c(x, t)u − u =0, (x, t) ∈ D, (1.1) 1 t u(x, 0) = u (x),x ∈ R , (1.2) where N N Lu = a (x, t)u , (b, ∇u)= b (x, t)u . (1.3) ik x x i x i i i,k=1 i=1 We assume that the following conditions hold. 1. The coeﬃcients of Equation (1.1) are continuous and satisfy the H¨older condition in each bounded subdomain Q of D, a = a (i, k =1,... ,N), and there are constants k > 0and ik
Journal of Mathematical Sciences – Springer Journals
Published: Jun 2, 2018
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