Solvability of a semilinear parabolic equation on Riemannian manifolds

Xinran Wei () and Mengmeng Zhang ()
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Xinran Wei: Central China Normal University
Mengmeng Zhang: Central China Normal University

Partial Differential Equations and Applications, 2024, vol. 5, issue 6, 1-22

Abstract: Abstract In this paper, we mainly consider the solvability of the initial value problem for the semilinear parabolic equation $$u_t-\triangle u+u=u^p$$ u t - ▵ u + u = u p in a Riemannian manifold M with a nonnegative Radon measure $$\mu $$ μ on M as initial data. When M is a N-dimensional connected and complete Riemannian manifold without boundary, and has bounded sectional curvature and positive injectivity radius, Takahashi and Yamamoto (J Evol Equ 23:55, 2023) have already studied the solvability of semilinear heat equation $$u_t-\triangle u=u^p$$ u t - ▵ u = u p . We use a substitution $$u=e^{-t}v$$ u = e - t v , then we can transform the semilinear parabolic equation $$u_t-\triangle u+u=u^p$$ u t - ▵ u + u = u p to the semilinear heat equation $$v_t-\triangle v=e^{-(p-1)t}v^p$$ v t - ▵ v = e - ( p - 1 ) t v p . Finally we get the sharp conditions and necessary conditions on the local-in-time solvability of the semilinear parabolic equation by using the properties of the heat kernel.

Keywords: Semilinear parabolic equation; Existence; Nonexistence; Riemannian manifold; 35K05; 35K08; 35K58 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s42985-024-00304-z

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