{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:Q4NHW6UVZ7LIAAQEWITBHE2SKV","short_pith_number":"pith:Q4NHW6UV","schema_version":"1.0","canonical_sha256":"871a7b7a95cfd6800204b2261393525575f390bdf14014b8103dd120981577de","source":{"kind":"arxiv","id":"1812.01933","version":3},"attestation_state":"computed","paper":{"title":"Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Ruzhansky, Nurgissa Yessirkegenov","submitted_at":"2018-12-05T11:57:35Z","abstract_excerpt":"In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold $M$: \\begin{equation*} \\begin{cases}\n u_{t}-\\mathfrak{L}_{M} u=f(u), \\;x\\in M, \\;t>0,\n \\\\u(0,x)=u_{0}(x), \\;x\\in M,\n \\end{cases} \\end{equation*} for $u_{0}\\geq 0$, where $\\mathfrak{L}_{M}$ is a sub-Laplacian of $M$. In the case when $M$ is a connected unimodular Lie group $\\mathbb G$, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with $u_{0}\\not\\equiv 0$, blow up in finite time if and only if $10,\n \\\\u(0,x)=u_{0}(x), \\;x\\in M,\n \\end{cases} \\end{equation*} for $u_{0}\\geq 0$, where $\\mathfrak{L}_{M}$ is a sub-Laplacian of $M$. In the case when $M$ is a connected unimodular Lie group $\\mathbb G$, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with $u_{0}\\not\\equiv 0$, blow up in finite time if and only if $1