On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term
For the semilinear dissipative hyperbolic Cauchy problem \[ u_{tt}- \Delta u+ f(u)= 0\text{ in } \mathbb{R}^N\times [0, \infty),\;u(t= 0)= u_0,\;u_t(t= 0)= u_1, \] where \(f\) satisfies assumptions like \(f(u)\cdot u\geq k_1 2 \int^u_0 f(s) ds\geq 0\) or \(|f(u)|\leq k_2|u|^{\alpha+ 1}\), \((k, \alpha> 0)\), detailed decay estimates of the solution are proved for the cases \(1\leq N\leq 6\), \(0< \alpha< 4/(N- 2)\). Both, energy methods and \(L^p- L^q\)-estimates for the linearized problem, are used.
scientific article; zbMATH DE number 894102