{"paper":{"title":"Stability of solutions to some evolution problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.DS","authors_text":"A.G.Ramm","submitted_at":"2010-12-13T16:08:01Z","abstract_excerpt":"Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\\dot{u}=A(t)u+F(t,u)+b(t), \\quad t\\ge 0; \\quad u(0)=u_0. \\qquad (*)$$ Here $\\dot{u}:=\\frac {du}{dt}$, $u=u(t)\\in H$, $t\\in \\R_+:=[0,\\infty)$, $A(t)$ is a linear dissipative operator: Re$(A(t)u,u)\\le -\\gamma(t)(u,u)$, $\\gamma(t)\\ge 0$, $F(t,u)$ is a nonlinear operator, $\\|F(t,u)\\|\\le c_0\\|u\\|^p$, $p>1$, $c_0,p$ are constants, $\\|b(t)\\|\\le \\beta(t),$ $\\beta(t)\\ge 0$ is a continuous function. Sufficient conditions are given for the solution $u(t)$ to problem (*) to exist for a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2785","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"}