Global infinite energy solutions of the critical semilinear wave equation
type: Item
instance of: scholarly article
title: Global infinite energy solutions of the critical semilinear wave equation
zbMATH Open document ID: 1173.35084
DOI: 10.4171/RMI/543
author: P. Germain
published in: Revista Matemática Iberoamericana
publication date: 2008-11-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1218475350
full work available at URL: https://eudml.org/doc/42381
review text: The author considers the Cauchy problem in \(\mathbb R^d\) for the semilinear wave equation \[ (\partial^2_t-\Delta)u +| u| ^{p-1}=0, \quad u_{t=0}=u_0 \quad\text{and}\quad \partial_t u_{t=0}=u_1 \] with critical exponent \(p=\frac{2d}{2-d} -1\). Using the method of Bourgain he proves that there exists a global solution if the initial data are of the form \((u_o,u_1)= (v_o+w_0, v_1+w_1)\) with \(E=\| v_o| \dot{H}_1\| + \| v_1| L_2 \| <\infty\) and \(\| w_o| \dot{B}^1_{2,\infty}\| + \| v_1| \dot{B}^0_{2,\infty} \| < C \exp(-\exp E^\kappa)\), \(\kappa,C >0\), \(d=3,4,6\). Moreover it is proved that there exists a global solution if the initial data are of the form \((u_o,u_1)= (v_o+c_0/| x| ^2, v_1+ c_1/| x| ^3)\) with \(\| v_o| \dot{H}_1\| + \| v_1| L_2 \| <\infty\), \(\max(c_0,c_0)<\varepsilon\) and \(d=6\). The last statement is proved by the Calderón method.
reviewed by: Leszek Skrzypczak
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 5361870
zbMATH Keywords: Besov spaces
zbMATH Keywords: method of Bourgain
zbMATH Keywords: Calderón method
MaRDI profile type: Publication
cites work: High Frequency Approximation of Solutions to Critical Nonlinear Wave Equations
cites work: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires
cites work: Q4255122
cites work: Existence of Weak Solutions for the Navier-Stokes Equations with Initial Data in L p
cites work: Flows of non-Lipschitzian vector fields and Navier-Stokes equations
cites work: On global infinite energy solutions to the Navier-Stokes equations in two dimensions
cites work: On global solutions to a defocusing semi-linear wave equation.
cites work: The global Cauchy problem for the nonlinear Klein-Gordon equation
cites work: Generalized Strichartz inequalities for the wave equation
cites work: Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity
cites work: Finite energy self–similar solutions of a nonlinear wave equation
cites work: Endpoint Strichartz estimates
cites work: Global well-posedness for semi-linear wave equations
cites work: Q4782579
cites work: On existence and scattering with minimal regularity for semilinear wave equations
cites work: Q4237139
cites work: Self-similar solutions and semi-linear wave equations in Besov spaces
cites work: Q4893982
cites work: The global Cauchy problem for a relativistic scalar field with power interaction
cites work: Regularity results for nonlinear wave equations
cites work: Q4317280
cites work: Geometric wave equations
cites work: A priori estimates for the wave equation and some applications
OpenAlex ID: W2071536554
description: scientific article; zbMATH DE number 5361870