Global solutions for dissipative Kirchhoff strings with m(r) = rp (p < 1)

[摘要] We investigate the evolution problem u+ deltau' + m(\A(1/2)u\(2)(H))Au = 0, u(0) = u(0), u'(0) = u(1), where H is a Hilbert space, A is a self-adjoint non-negative operator on H with domain D(A), delta > 0 is a parameter, and m(r) = r(p) with p < I. We prove that this problem has a unique global solution for positive times, provided that the initial data (u(0),u(1)) D(A(alphai/2)) X D(A((alphai-1)/2)) satisfy a suitable smallness assumption and the non-degeneracy condition m(\A(1/2)u(0)\(2)(H)) > 0 (where p greater than or equal to 2(-i) and alpha (i) = 2(i) + 1). Moreover, we prove for this solution decay with a polynomial rate as t --> +infinity These results apply to degenerate hyperbolic PDEs with non-local non-linearities. (C) 2000 Academic Press.

[发布日期] 2000-10-01

[关键词] hyperbolic equations;degenerate hyperbolic equations;dissipative equations;global existence;Kirchhoff equations;asymptotic behavior