{"paper":{"title":"Asymptotic stability of solutions to abstract differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"A.G.Ramm","submitted_at":"2010-09-30T13:28:07Z","abstract_excerpt":"An evolution problem for abstract differential equations is studied. The typical problem is: $$\\dot{u}=A(t)u+F(t,u), \\quad t\\geq 0; \\,\\, u(0)=u_0;\\quad \\dot{u}=\\frac {du}{dt}\\qquad (*)$$ Here $A(t)$ is a linear bounded operator in a Hilbert space $H$, and $F$ is a nonlinear operator, $\\|F(t,u)\\|\\leq c_0\\|u\\|^p,\\,\\,p>1$, $c_0, p=const>0$. It is assumed that Re$(A(t)u,u)\\leq -\\gamma(t)\\|u\\|^2$ $\\forall u\\in H$, where $\\gamma(t)>0$, and the case when $\\lim_{t\\to \\infty}\\gamma(t)=0$ is also considered. An estimate of the rate of decay of solutions to problem (*) is given. The derivation of this es"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.6124","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}