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On the two critical exponents for a dispersive equation with weak dissipation
Abstract
In this paper, we study a nonlinear Cauchy problem written in normal form utt + A u+ But =D|u|α, & t≥ 0, x ∈ ℝn u(0,x)=u0(x), ut(0,x)=u1(x). Here~ Af=ℱ−1(a(ξ)2ℱf), with a(ξ) homogeneous of order k > 0 under the assumption that the Hessian matrix of a(ξ) is nonsingular, the operator Bf=ℱ−1(b(ξ)ℱf) represents a weak dissipation, in the sense that b(ξ) ∼ 1 as |ξ| → ∞ and %, in several models, D=B but we also consider the case Df=ν∗ (−Δ)δ(I−Δ)−δ, with ν a bounded measure and δ≥ k/2. We derive Lp−Lq estimates of the linear solution operator to the considered model and we prove the transition from a Fujita-type exponent to a Strauss-type exponent for the global-in-time existence of small data solution for the corresponding nonlinear problems.
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BibTeX
@misc{dabbicco2026critical,
title = {{On the two critical exponents for a dispersive equation with weak dissipation}},
author = {Marcello D'Abbicco and Antonio Lagioia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22616},
eprint = {2610.22616},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22616},
}
RIS
TY - UNPB
TI - On the two critical exponents for a dispersive equation with weak dissipation
AU - Marcello D'Abbicco
AU - Antonio Lagioia
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22616
UR - https://arxiv.org/abs/2610.22616
AB - In this paper, we study a nonlinear Cauchy problem written in normal form u_tt + A u+ Bu_t =D|u|^α, & t≥ 0, x ∈ ℝ^n u(0,x)=u_0(x), u_t(0,x)=u_1(x). Here~ Af=ℱ^−1(a(ξ)^2ℱf), with a(ξ) homogeneous of order k > 0 under the assumption that the Hessian matrix of a(ξ) is nonsingular, the operator Bf=ℱ^−1(b(ξ)ℱf) represents a weak dissipation, in the sense that b(ξ) ∼ 1 as |ξ| → ∞ and %, in several models, D=B but we also consider the case Df=ν∗ (−Δ)^δ(I−Δ)^−δ, with ν a bounded measure and δ≥ k/2. We derive L^p−L^q estimates of the linear solution operator to the considered model and we prove the transition from a Fujita-type exponent to a Strauss-type exponent for the global-in-time existence of small data solution for the corresponding nonlinear problems.
ER -