{"paper":{"title":"Perturbations of self-similar solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fl\\'avio Dickstein, Fred B. Weissler, Ivan Naumkin, Thierry Cazenave","submitted_at":"2018-05-11T16:07:09Z","abstract_excerpt":"We consider the nonlinear heat equation $u_t = \\Delta u + |u|^\\alpha u$ with $\\alpha >0$, either on ${\\mathbb R}^N $, $N\\ge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) \\alpha <4$, for every $\\mu \\in {\\mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = \\mu |x-x_0|^{-\\frac {2} {\\alpha }}$ in a neighborhood of some $x_0\\in \\Omega $ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.04466","kind":"arxiv","version":2},"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"}