Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations (Q1323446)

We consider the Cauchy problem for the semilinear wave equation with a dissipative term: \[ u_{tt}- \Delta u+ u_ t- | u|^ p u =0\quad\text{ in } \mathbb{R}^ N\times [0,\infty), \qquad u(x,0)=u_ 0(x), \quad u_ t(x,0)=u_ 1(x). \tag{*} \] By use of ``potential well'' or ``modified potential well'' method, we prove the global existence and decay property of weak solutions for the equation (*) with \(4/N\leq p<4/(N-2)\) under the assumption that \(\| u_ 1\|^ 2+ \|\nabla u_ 0\|^ 2\) is small. For our results, the dissipative term \(u_ t\) plays an essential role. Moreover, by applying the above method, we discuss on the life span of weak solutions for both of the equations with and without dissipative term.