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Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. (English) Zbl 1029.35180
Summary: We study the global existence and asymptotic behavior of solutions to the Cauchy problem for the semilinear dissipative wave equations: $\square u+\partial_t u=|u|^{\alpha+1}$, $u|_{t=0}= \varepsilon u_0\in H^1\cap L^1$, $\partial_tu|_{t=0} =\varepsilon u_1\in L^2 \cap L^1$ with a small parameter $\varepsilon>0$. When $N\le 3$ and $2/N< \alpha \le 2/[N-2]^+$, we show the global solvability and derive the sharp rates of the solutions.

35L70Nonlinear second-order hyperbolic equations
35B40Asymptotic behavior of solutions of PDE
35L15Second order hyperbolic equations, initial value problems
35B33Critical exponents (PDE)
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