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Asymptotic Behaviour of Solutions to some Pseudoparabolic Equations
Author(s) -
Karch Grzegorz
Publication year - 1997
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(199702)20:3<271::aid-mma859>3.0.co;2-f
Subject(s) - mathematics , heat equation , initial value problem , cauchy problem , constant (computer programming) , kernel (algebra) , mathematical physics , heat kernel , mathematical analysis , pure mathematics , computer science , programming language
The aim of this paper is to investigate the behaviour as t →∞ of solutions to the Cauchy problem u t −Δ u t −νΔ u −( b , ∇ u )=∇⋅ F ( u ), u ( x , 0)= u 0 ( x ), where ν>0 is a fixed constant, t ⩾0, x ∈ℝ n . First, we prove that if u is the solution to the linearized equation, i.e. with ∇⋅ F ( u )≡0, then u decays like a solution for the analogous problem to the heat equation. Moreover, the long‐time behaviour of u is described by the heat kernel. Next, analogous results are established for the non‐linear equation with some assumptions imposed on F, p , and the initial condition u 0 . © 1997 by B.G. Teubner Stuttgart‐John Wiley & Sons, Ltd.