Existence of backward global-solutions to nonlinear dissipative wave-equations

Abstract

Let OMEGA be a bounded smooth domain of R(n). We prove existence of global solutions, i. e. solutions defined for all t is-an-element-of R, for dissipative wave equations of the form: u''-DELTAu+\u'\p-1 u'=0 in (- infinity, infinity) x OMEGA with Dirichlet homogeneous boundary conditions, where 1 < p < infinity if n less-than-or-equal-to 2 or 1 < p less-than-or-equal-to (n + 2)/(n - 2) if n > 2. More precisely, for every solution psi (with constant sign if 1 < p < 2) of an elliptic problem we prove the existence of a solution growing like \t\(p/(p-1)) when t --> - infinity. When OMEGA is unbounded the same existence result holds for p greater-than-or-equal-to 2.