Nonlinear Neumann boundary value problem for semilinear heat equations with critical power nonlinearities

Article type: Research Article

Authors: Hayashi, Nakao^{a; *} | Kaikina, Elena I.^b | Naumkin, Pavel I.^b | Ogawa, Takayoshi^c

Affiliations: [a] Research Alliance Center for Mathematical Sciences, Tohoku University, Sendai 980-8578, Japan. E-mail: [email protected] | [b] Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico. E-mails: [email protected], [email protected] | [c] Mathematical Institute, Tohoku University, Sendai 980-8578, Japan. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We study the nonlinear Neumann boundary value problem for semilinear heat equation ∂tu−Δu=λ|u|p,t>0,x∈R+n,u(0,x)=εu0(x),x∈R+n,−∂xu(t,x′,0)=γ|u|q(t,x′,0),t>0,x′∈Rn−1 where p=1+2n, q=1+1n and ε>0 is small enough. We investigate the life span of solutions for λ,γ>0. Also we study the global in time existence and large time asymptotic behavior of solutions in the case of λ,γ<0 and ∫R+nu0(x)dx>0.

Keywords: Nonlinear heat equation, large time asymptotics, initial-boundary value problem, half line

DOI: 10.3233/ASY-211751

Journal: Asymptotic Analysis, vol. 130, no. 1-2, pp. 261-295, 2022