On the wellposed Cauchy problem for some dispersive equations
type: Item
instance of: scholarly article
title: On the wellposed Cauchy problem for some dispersive equations
zbMATH Open document ID: 0851.35076
author: Shigeo Tarama
published in: Journal of the Mathematical Society of Japan
publication date: 1996-11-18T00:00:00Z
review text: The forward Cauchy problem for the degenerate parabolic operator with real coefficients \[ \partial_t u(t, x)+ a(x) \partial^2_x u(t, x)+ b(x) \partial_x u(t, x)+ c(x) u(t, x)= 0 \] is \(L^2\)- and \(H^\infty\)-well posed if and only if \(a(x)\leq 0\). The question which is studied in the present paper is what happens with the forward Cauchy problem if we add to the operator the dispersive term \(\partial^3_x u(t, x)\). The author shows that the condition \[ \int^y_x a(s) ds\leq K\;\Biggl( \int^y_x a(s) ds= K\log (y- x+ 2)\Biggr),\quad x\leq y \] is necessary and sufficient for the \(L_2\)-well posedness (\(H^\infty\)-well posedness). The necessity will be proved by the construction of asymptotic solutions. For the proof of the sufficiency, the author uses the fact that the Cauchy problem for the equation \[ \partial_t u(t, x)+ \partial^3_x u(t, x)+ a(x) \partial^2_x u(t, x)+ b(x) \partial_x u(t, x)+ c(x) u(t, x)= 0 \] is \(L^2\)-, \(H^\infty\)- and \(S\)-well-posed if \(a(x)\leq 0\). Under the above condition a suitable transformation transfers the original problem to the last one.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35G10
zbMATH DE Number: 844446
zbMATH Keywords: well posedness
zbMATH Keywords: forward Cauchy problem
zbMATH Keywords: dispersive term
reviewed by: Michael Reissig
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04710143
OpenAlex ID: W2057461994
DOI: 10.2969/JMSJ/04710143
description: scientific article; zbMATH DE number 844446