Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups
Ruzhansky M. Yessirkegenov N.
25 January 2022Academic Press Inc.
Journal of Differential Equations
2022#308455 - 473 pp.
In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold M: {ut−LMu=f(u),x∈M,t>0,u(0,x)=u0(x),x∈M, for u0≥0, where LM is a sub-Laplacian of M. In the case when M is a connected unimodular Lie group G, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with u0≢0, blow up in finite time if and only if 1F:=1+2/D when f(u)≃up, where D is the global dimension of G. In the case 1F and when f:[0,∞)→[0,∞) is a locally integrable function such that f(u)≥K2up for some K2>0, we also show that the differential inequality ut−LMu≥f(u) does not admit any nontrivial distributional (a function u∈Llocp(Q) which satisfies the differential inequality in D′(Q)) solution u≥0 in Q:=(0,∞)×G. Furthermore, in the case when G has exponential volume growth and f:[0,∞)→[0,∞) is a continuous increasing function such that f(u)≤K1up for some K1>0, we prove that the Cauchy problem has a global, classical solution for 10∈Lq(G) with 1≤q<∞. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds M.
Differential inequality , Global well-posedness , Semilinear heat equation , Sub-Laplacian , Sub-Riemannian manifold , Unimodular Lie group
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Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Belgium
School of Mathematical Sciences, Queen Mary University of London, United Kingdom
Suleyman Demirel University, Kaskelen, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Kazakhstan
Department of Mathematics: Analysis
School of Mathematical Sciences
Suleyman Demirel University
Institute of Mathematics and Mathematical Modeling
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