Semilinear Equations Including the Mixed Operator

2025/02/23 by Ayoub, Alaa
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2502.16646

openalex publication_date 2025/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the local and global existence of solutions to a semilinear evolution equation driven by a mixed local-nonlocal operator of the form \( L = -Δ+ (-Δ)^α/2 \), where \( 0 < α< 2 \). The Cauchy problem under consideration is ∂_t u + t^βL u = -h(t) u^p, x ∈ ℝ^N, t gt; 0, with nonnegative initial data \( u(x, 0) = u₀(x) \). We establish the existence and uniqueness of local solutions in \( L^∞(ℝ^N) \) using a contraction mapping argument. Furthermore, we analyze conditions for global existence, proving that solutions remain globally bounded in time under appropriate assumptions on the parameters \( β\), \( p \), and the function \( h(t) \).

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