Damped wave equation with a critical nonlinearity

Abstract

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity { ∂ t 2 u + ∂ t u − Δ u + λ u 1 + 2 n = 0 , x ∈ R n , t > 0 , u ( 0 , x ) = ε u 0 ( x ) , ∂ t u ( 0 , x ) = ε u 1 ( x ) , x ∈ R n , \begin{equation*} \left \{ \begin {array}{c} \partial _{t}^{2}u+\partial _{t}u-\Delta u+\lambda u^{1+\frac {2}{n}}=0,\text { }x\in \mathbf {R}^{n},\text { }t>0, u(0,x)=\varepsilon u_{0}\left ( x\right ) ,\partial _{t}u(0,x)=\varepsilon u_{1}\left ( x\right ) ,x\in \mathbf {R}^{n}, \end{array} \right . \end{equation*} where ε > 0 , \varepsilon >0, and space dimensions n = 1 , 2 , 3 n=1,2,3 . Assume that the initial data u 0 ∈ H δ , 0 ∩ H 0 , δ , u 1 ∈ H δ − 1 , 0 ∩ H − 1 , δ , \begin{equation*} u_{0}\in \mathbf {H}^{\delta ,0}\cap \mathbf {H}^{0,\delta },\text { }u_{1}\in \mathbf {H}^{\delta -1,0}\cap \mathbf {H}^{-1,\delta }, \end{equation*} where δ > n 2 , \delta >\frac {n}{2}, weighted Sobolev spaces are H l , m = { ϕ ∈ L 2 ; ‖ ⟨ x ⟩ m ⟨ i ∂ x ⟩ l ϕ ( x ) ‖ L 2 > ∞ } , \begin{equation*} \mathbf {H}^{l,m}=\left \{ \phi \in \mathbf {L}^{2};\left \Vert \left \langle x\right \rangle ^{m}\left \langle i\partial _{x}\right \rangle ^{l}\phi \left ( x\right ) \right \Vert _{\mathbf {L}^{2}}>\infty \right \} , \end{equation*} ⟨ x ⟩ = 1 + x 2 . \left \langle x\right \rangle =\sqrt {1+x^{2}}. Also we suppose that λ θ 2 n > 0 , ∫ u 0 ( x ) d x > 0 , \begin{equation*} \lambda \theta ^{\frac {2}{n}}>0,\int u_{0}\left ( x\right ) dx>0, \end{equation*} where θ = ∫ ( u 0 ( x ) + u 1 ( x ) ) d x . \begin{equation*} \text { }\theta =\int \left ( u_{0}\left ( x\right ) +u_{1}\left ( x\right ) \right ) dx\text {.} \end{equation*} Then we prove that there exists a positive ε 0 \varepsilon _{0} such that the Cauchy problem above has a unique global solution u ∈ C ( [ 0 , ∞ ) ; H δ , 0 ) u\in \mathbf {C}\left ( \left [ 0,\infty \right ) ;\mathbf {H}^{\delta ,0}\right ) satisfying the time decay property ‖ u ( t ) − ε θ G ( t , x ) e − φ ( t ) ‖ L p ≤ C ε 1 + 2 n g − 1 − n 2 ( t ) ⟨ t ⟩ − n 2 ( 1 − 1 p ) \begin{equation*} \left \Vert u\left ( t\right ) -\varepsilon \theta G\left ( t,x\right ) e^{-\varphi \left ( t\right ) }\right \Vert _{\mathbf {L}^{p}}\leq C\varepsilon ^{1+\frac {2}{n}}g^{-1-\frac {n}{2}}\left ( t\right ) \left \langle t\right \rangle ^{-\frac {n}{2}\left ( 1-\frac {1}{p}\right ) } \end{equation*} for all t > 0 , t>0, 1 ≤ p ≤ ∞ , 1\leq p\leq \infty , where ε ∈ ( 0 , ε 0 ] . \varepsilon \in \left ( 0,\varepsilon _{0}\right ] .