L^p-L^q estimates for damped wave equations and their applications to semi-linear problem

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概要

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• In this paper we study the Cauchy problem to the linear damped wave equation u<SUB>tt</SUB>-Δ u+2au<SUB>t</SUB>=0 in (0, ∞)× \bm{R}<SUP>n</SUP> (n≥q 2). It has been asserted that the above equation has the diffusive structure as t→∞. We give the precise interpolation of the diffusive structure, which is shown by L^{p<SUB>-</SUB>}L<SUP>q</SUP> estimates. We apply the above L^{p<SUB>-</SUB>}L<SUP>q</SUP> estimates to the Cauchy problem for the semilinear damped wave equation u<SUB>tt</SUB>-Δ u+ 2au<SUB>t</SUB>=|u|<SUP>σ</SUP>u in (0, ∞)× \bm{R}<SUP>n</SUP> (2≤ n≤ 5). If the power σ is larger than the critical exponent 2/n (Fujita critical exponent) and it satisfies σ≤ 2/(n-2) when n≥q 3, then the time global existence of small solution is proved, and the decay estimates of several norms of the solution are derived.

• 社団法人 日本数学会の論文
• 2004-04-01

著者

• Narazaki Takashi

  Department Of Mathematical Sciences Faculty Of Science Tokai University

関連論文

• On the Existence of Classical periodic Solutions of the Equation u_tt-(σ(u_x))_x-u_xxt=f
• L^p-L^q estimates for damped wave equations and their applications to semi-linear problem
• On Some Nonlinear Evolution Equations in Non-cylindrical Domain
• Existence and Decay of Classical Solutions of Some Nonlinear Evolution Equations with the Strong Dissipation

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