Entire solutions with moving singularities for a semilinear heat equation with a critical exponent

Jin Takahashi ()
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Jin Takahashi: Tokyo Institute of Technology

Partial Differential Equations and Applications, 2022, vol. 3, issue 2, 1-16

Abstract: Abstract We consider the semilinear heat equation $$\partial _t u -\Delta u=u^{N/(N-2)}$$ ∂ t u - Δ u = u N / ( N - 2 ) in $$\Omega $$ Ω with $$u=0$$ u = 0 on $$\partial \Omega $$ ∂ Ω , where $$N\ge 3$$ N ≥ 3 and $$\Omega $$ Ω is a smooth bounded domain in $$\mathbf {R}^N$$ R N . Let $$\xi :\mathbf {R}\rightarrow \Omega $$ ξ : R → Ω satisfy $$\overline{\{\xi (t);t\in \mathbf {R}\}}\subset \Omega $$ { ξ ( t ) ; t ∈ R } ¯ ⊂ Ω . Under some assumption on the uniform Hölder continuity of $$\xi $$ ξ , we construct a nonnegative solution u defined for all $$t\in \mathbf {R}$$ t ∈ R satisfying $$u(x,t)\rightarrow \infty $$ u ( x , t ) → ∞ for each $$t\in \mathbf {R}$$ t ∈ R as $$x\rightarrow \xi (t)$$ x → ξ ( t ) .

Keywords: Semilinear heat equation; Moving singularity; Entire solution; Primary 35K58; Secondary 35A01; 35A21; 35B08 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-022-00164-5

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