On solvability of initial-boundary value problems for a nonlinear equation of variable type
type: Item
instance of: scholarly article
title: On solvability of initial-boundary value problems for a nonlinear equation of variable type
zbMATH Open document ID: 0917.35080
DOI: 10.1007/BF02674123
author: S. N. Glazatov
published in: Siberian Mathematical Journal
publication date: 1999-05-10T00:00:00Z
review text: The author studies solvability of the following boundary value problems: find a solution to the equation \[ uu_t-u_{xx}=f(x,t)\quad (0<x<l,\;0<t<T) \tag{1} \] satisfying the conditions \[ u|_{t=0}=u_0(x),\quad u|_{x=0}=\varphi_1(t),\quad u|_{x=l}=\varphi_2(t) \tag{2} \] or \[ u|_{t=0}=u_0(x),\quad u_x|_{x=0}=u_x|_{x=l}=0. \tag{3} \] Note that (1) is a nonlinear equation with variable time direction. The author proves that the boundary value problem (1), (2) has a unique regular solution provided that, in addition to some smoothness and agreement conditions, the following conditions hold: (i) \(u_0(x) \geq \delta > 0\), \(x\in[0,l]\), \(\varphi_i(t)\geq\delta>0\), \(t\in [0,T]\); (ii) \(l\) is small; or, alternatively, (i) and the condition (iii) the functions \(f(x,t)\), \(u_0(x)\), and \(\varphi_i(t)\) are small. The boundary value problem (1), (3) has a unique regular solution provided that, in addition to some smoothness and agreement conditions, the following conditions hold: (i\('\)) \(u_0(x)\geq\delta>0\), \(x\in[0,l]\); (iii\('\)) the functions \(f(x,t)\) and \(u_0(x)\) are small. The above smallness conditions on \(l\) or \(f(x,t)\), \(u_0(x)\), \(\varphi_1(t)\), \(\varphi_2(t)\) are found. Note that, unlike many preceding papers, the condition \(f(x,t)\geq 0\), guaranteeing that the solution preserves sign, is omitted.
Mathematics Subject Classification ID: 35M10
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 1285462
zbMATH Keywords: smallness conditions
zbMATH Keywords: nondefinite solution
zbMATH Keywords: solvability
zbMATH Keywords: variable time direction
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1285462