{"paper":{"title":"Dimension of attractors and invariant sets in reaction diffusion equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Martino Prizzi","submitted_at":"2011-02-20T10:47:19Z","abstract_excerpt":"Under fairly general assumptions, we prove that every compact invariant set $\\mathcal I$ of the semiflow generated by the semilinear reaction diffusion equation  u_t+\\beta(x)u-\\Delta u&=f(x,u),&&(t,x)\\in[0,+\\infty[\\times\\Omega, u&=0,&&(t,x)\\in[0,+\\infty[\\times\\partial\\Omega} {equation*} in $H^1_0(\\Omega)$ has finite Hausdorff dimension. Here $\\Omega$ is an arbitrary, possibly unbounded, domain in $\\R^3$ and $f(x,u)$ is a nonlinearity of subcritical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\\mathcal I$ needs not to be an an a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4062","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"}