<p>The topic of this paper is a defocusing semi-linear wave equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(u_{t t}-\varDelta u=-|u|^{p-1} u\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">tt</mi> </mrow> </msub> <mo>-</mo> <mi>Δ</mi> <mi>u</mi> <mo>=</mo> <mo>-</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> in sub-conformal case in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(d\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> whose initial data come with a finite energy. We prove that almost all energy moves to infinity at almost the light speed as time tends to infinity. In addition, the inward/outward part of energy gradually vanishes as time tends to positive/negative infinity. We also prove some decay estimates of the solutions if initial data decay at a certain rate as the spatial variable tends to infinity. A combination of this property with a method of characteristic lines give a scattering result if the initial data are radial and in a weighted energy space.</p>

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Energy distribution of solutions to defocusing semi-linear wave equation in \(\mathbf {d\ge 3}\)

  • Liang Li,
  • Ruipeng Shen

摘要

The topic of this paper is a defocusing semi-linear wave equation \(u_{t t}-\varDelta u=-|u|^{p-1} u\) u tt - Δ u = - | u | p - 1 u in sub-conformal case in \(d\ge 3\) d 3 whose initial data come with a finite energy. We prove that almost all energy moves to infinity at almost the light speed as time tends to infinity. In addition, the inward/outward part of energy gradually vanishes as time tends to positive/negative infinity. We also prove some decay estimates of the solutions if initial data decay at a certain rate as the spatial variable tends to infinity. A combination of this property with a method of characteristic lines give a scattering result if the initial data are radial and in a weighted energy space.