Extinction in a finite time for solutions of a class of quasilinear parabolic equations

Article type: Research Article

Authors: Belaud, Y.^{a; b; *} | Shishkov, A.^{c; d}

Affiliations: [a] Laboratoire de Mathématiques et Physique Théorique, Université de Tours, UFR Sciences et Techniques, Parc de Grandmont, 37200 Tours, France | [b] Fédération de recherche Denis Poisson, Laboratoire MAPMO rue de Chartres Bât. Maths., 45067 Orleans Cedex 2, France | [c] Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Dobrovol’skogo str. 1, Slaviansk, 84116, Ukraine | [d] Peoples’ Friendship University of Russia, Miklukho-Maklaya str. 6, Moscow, 117198, Russia

Correspondence: [*] Corresponding author. E-mail: yves.belaud@lmpt.univ-tours.fr.

Abstract: We study the property of extinction in a finite time for nonnegative solutions of 1q∂∂t(uq)−∇(|∇u|p−2∇u)+a(x)uλ=0 for the Dirichlet Boundary Conditions when q>λ>0, p⩾1+q, p⩾2, a(x)⩾0 and Ω a bounded domain of RN (N⩾1). We prove some necessary and sufficient conditions. The threshold is for power functions when p>1+q while finite time extinction occurs for very flat potentials a(x) when p=1+q.

Keywords: Extinction time, semi-classical limit, nonlinear eigenvalue problem

DOI: 10.3233/ASY-211674

Journal: Asymptotic Analysis, vol. 127, no. 1-2, pp. 97-119, 2022