scholarly journals Global solutions and self-similar solutions for nonlinear beam equations

2006 ◽  
Vol 44 (3-4) ◽  
pp. 405-412 ◽  
Author(s):  
Ai Guo ◽  
Shangbin Cui
Keyword(s):  
Global Solutions ◽  
Nonlinear Beam ◽  
Beam Equations ◽  
Self Similar ◽  
Similar Solutions

ASYMPTOTICALLY SELF-SIMILAR GLOBAL SOLUTIONS OF A DAMPED WAVE EQUATION

2007 ◽  
Vol 09 (02) ◽  
pp. 253-277 ◽  
Author(s):  
SEIFEDDINE SNOUSSI ◽  
SLIM TAYACHI
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Global Solutions ◽  
The Self ◽  
Initial Condition ◽  
Damped Wave Equation ◽  
Small Initial Data ◽  
Additional Hypothesis ◽  
Self Similar ◽  
Similar Solutions

We study the existence and the asymptotic behavior of global solutions of the damped wave equation [Formula: see text] where a ∈ ℝ, α >1, t > 0, x ∈ ℝn, n = 1,2,3, with initial condition (u (0), ut (0)) = (φ,ψ). For α > 2 and α > 1+2 / n, we prove the existence of mild global solutions for small initial data with low regularity and which are not in L1(ℝn). Under the additional hypothesis, (2 < α < 5, when n = 3), we prove that some of these solutions are asymptotic to the self-similar solutions of the associated semi-linear heat equation [Formula: see text] with homogeneous slowly decreasing initial data behaving like c|x|-2 / (α-1) as |x|→ ∞.

Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data

2019 ◽  
Vol 150 (2) ◽  
pp. 789-811
Author(s):  
Yūki Naito
Keyword(s):  
Initial Data ◽  
Existence Of Solutions ◽  
Global Solutions ◽  
Heat Equations ◽  
Complicated Manner ◽  
The Cauchy Problem ◽  
Self Similar ◽  
The One ◽  
Image Position ◽  
Similar Solutions

AbstractWe consider the Cauchy problem $$\left\{ {\matrix{ {u_t = \Delta u + u^p,\quad } \hfill & {x\in {\bf R}^N,\;t \leq 0,} \hfill \cr {u(x,0) = u_0(x),\quad } \hfill & {x\in {\bf R}^N,} \hfill \cr } } \right.$$where N > 2, p > 1, and u0 is a bounded continuous non-negative function in RN. We study the case where u0(x) decays at the rate |x|−2/(p−1) as |x| → ∞, and investigate the convergence property of the global solutions to the forward self-similar solutions. We first give the precise description of the relationship between the spatial decay of initial data and the large time behaviour of solutions, and then we show the existence of solutions with a time decay rate slower than the one of self-similar solutions. We also show the existence of solutions that behave in a complicated manner.

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