In this note we show that all small solutions in the energy space of the generalized 1D Boussinesq equation must decay to zero as time tends to infinity, strongly on slightly proper subsets of the space-time light cone. Our result does not require any assumption on the power of the nonlinearity, working even for the supercritical range of scattering. For the proof, we use two new Virial identities in the spirit of works [10,11]. No parity assumption on the initial data is needed.
Leta∈Lloc1(ℝ+)andk∈C(ℝ+)be given. In this paper, we study the functional equationR(s)(a*R)(t)-(a*R)(s)R(t)=k(s)(a*R)(t)-k(t)(a*R)(s), for bounded operator valued functionsR(t)defined on the positive real lineℝ+. We show that, under some natural assumptions ona(·)andk(·), every solution of the above mentioned functional equation gives rise to a commutative(a,k)-resolvent familyR(t)generated byAx=lim t→0+(R(t)x-k(t)x/(a*k)(t))defined on the domainD(A):={x∈X:lim t→0+(R(t)x-k(t)x/(a*k)(t))exists inX}and, conversely, that each(a,k)-resolvent familyR(t)satisfy the above mentioned functional equation. In particular, our study produces new functional equations that characterize semigroups, cosine operator families, and a class of operator families in between them that, in turn, are in one to one correspondence with the well-posedness of abstract fractional Cauchy problems.
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