Abstract
We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{@{}ll} u_{tt} - \Delta u + b(t)u_t=f(u)\,, & (t,x) \in {\bf R}^+ \times {\bf R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x)\,, & x \in {\bf R}^N\,. \end{array}\right. {(*)} $$ hen $b(t)=(t+1)^{-\beta}$ with $0\le \beta <1$, the damping is effective and the solution $u$ to ($*$) behaves as that to the corresponding parabolic problem. When $f(u)=O(|u|^{\rho})$ as $u \to 0$ with $1<\rho < \frac{N+2}{[N-2]_+}$(the Sobolev exponent), our main aim is to show the time-global existence of solutions for small data in the supercritical exponent $\rho>\rho_F(N):=1+2/N$. We also obtain some blow-up results on the solution within a finite time, so that the smallness of the data is essential to get global existence in the supercritical exponent case.
Citation
Kenji NISHIHARA. "Asymptotic Behavior of Solutions to the Semilinear Wave Equation with Time-dependent Damping." Tokyo J. Math. 34 (2) 327 - 343, December 2011. https://doi.org/10.3836/tjm/1327931389
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