POINTWISE ESTIMATES FOR THE WAVE EQUATION WITH DISSIPATION IN ODD SPATIAL DIMENSION

2008 ◽  
Vol 05 (02) ◽  
pp. 477-486 ◽  
Author(s):  
HONGMEI XU ◽  
WEIKE WANG
Keyword(s):  
Wave Equation ◽  
Large Time ◽  
Pointwise Estimate ◽  
Spatial Dimension ◽  
Large Time Behavior ◽  
Time Behavior ◽  
The Cauchy Problem ◽  
Explicit Analysis ◽  
Huygens Principle ◽  
The Green Function

We study the pointwise estimate of solution to the Cauchy problem for the wave equation with viscosity in odd spatial dimension. Through the explicit analysis of the Green function, we obtain the large time behavior of solution, and the solution exhibit the generalized Huygens principle.

scholarly journals SMOOTHING EFFECT AND LARGE TIME BEHAVIOR OF SOLUTIONS TO SCHRÖDINGER EQUATIONS WITH NONLINEARITY OF INTEGRAL TYPE

2004 ◽  
Vol 06 (04) ◽  
pp. 681-703 ◽  
Author(s):  
T. OZAWA ◽  
Y. YAMAZAKI
Keyword(s):  
Cauchy Problem ◽  
Large Time ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Smoothing Effect ◽  
Range Effect ◽  
Integral Type ◽  
Phase Modification ◽  
The Cauchy Problem ◽  
Two Space Dimensions

We study the smoothing effect in space and asymptotic behavior in time of solutions to the Cauchy problem for the nonlinear Schrödinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. A detailed description is given on the phase modification of scattering solutions by taking into account the long range effect of the interaction.

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