2025/02/28 by Kumar, Vishvesh, Torebek, Berikbol T. · 4 citations
#35A01 #35B33 #35B44 #35K58 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.21273
This paper explores the critical behavior of the semilinear heat equation ut+La, bu=|u|p+f(x), considering both the presence and absence of a forcing term f(x). The mixed local-nonlocal operator La, b=-aΔ+b(-Δ)s, a, b ∈ ℝ+, incorporates both local and nonlocal Laplacians. We determine the Fujita-type critical exponents by considering the existence or nonexistence of global solutions. Interestingly, the critical exponent is determined by the nonlocal component of the operator and, as a result, coincides with that of the fractional Laplacian. In the case without a forcing term, our results improve upon recent findings by Biagi et al. [Bull. London Math. Soc. 57 (2025), 265-284] and Del Pezzo et al. [Nonlinear Analysis 255 (2025), 113761]. When a forcing term is included, our results refine those of Wang et al. [J. Math. Anal. Appl., 488 (1) (2020), 124067] and complement the work of Majdoub [La Matematica, 2 (2023), 340-361].