{"paper":{"title":"Large time behaviour of higher dimensional logarithmic diffusion equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kin Ming Hui, SungHoon Kim","submitted_at":"2011-11-24T08:34:45Z","abstract_excerpt":"Let $n\\ge 3$ and $\\psi_{\\lambda_0}$ be the radially symmetric solution of $\\Delta\\log\\psi+2\\beta\\psi+\\beta x\\cdot\\nabla\\psi=0$ in $R^n$, $\\psi(0)=\\lambda_0$, for some constants $\\lambda_0>0$, $\\beta>0$. Suppose $u_0\\ge 0$ satisfies $u_0-\\psi_{\\lambda_0}\\in L^1(R^n)$ and $u_0(x)\\approx\\frac{2(n-2)}{\\beta}\\frac{\\log |x|}{|x|^2}$ as $|x|\\to\\infty$. We prove that the rescaled solution $\\widetilde{u}(x,t)=e^{2\\beta t}u(e^{\\beta t}x,t)$ of the maximal global solution $u$ of the equation $u_t=\\Delta\\log u$ in $R^n\\times (0,\\infty)$, $u(x,0)=u_0(x)$ in $R^n$, converges uniformly on every compact subse"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5692","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"}