On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian

Abstract

Abstract. Let Ω ⊂ R N be an arbitrary bounded open set. We consider a degenerate parabolic equation associated to the fractional p-Laplace operator (−∆) s p (p ≥ 2, s ∈ (0, 1)) with the Dirichlet boundary condition and a monotone perturbation growing like |τ | q−2 τ, q > p and with bad sign at infinity as |τ | → ∞. We show the existence of locally-defined strong solutions to the problem with any initial condition u 0 ∈ L r (Ω) where r ≥ 2 satisfies r > N (q − p)/sp. Then, we prove that finite time blow-up is possible for these problems in the range of parameters provided for r, p, q and the initial datum u 0 . Contents

Blow-up and global existence of solutions to a parabolic equation associated with the fraction <i>p</i>-Laplacian

“…In order to study the dynamics of the solutions to problem (1), as in [9], we define the following energy functional:…”

“…Fractional operators and their applications have been attracting considerable attention in the research field of analysis of nonlinear PDEs, we refer the readers to [3,5,6,7,20] and the references therein. Problem (1) was studied by Gal and Warm in [9], where the existence of locally-defined strong solutions and finite time blow-up of the strong solutions was considered. The local existence result can be stated as the following theorem (see […”

Optimal control of the coefficient for the regional fractional <inline-formula><tex-math id="M1">\begin{document} $p$\end{document}</tex-math></inline-formula>-Laplace equation: Approximation and convergence

“…The nonlocality of L s Ω,p (κ, •) makes both the state equation (12), and OCP extremely challenging. Indeed the papers [25,45], where the authors considered κ = 1, realized that the standard techniques available for the local p-Laplace equation are not directly applicable to the regional fractional p-Laplace equation (1.2). For OCP the additional complication occurs due to the fact that the operator L s Ω,p (κ, •) may degenerate, see Subsection 2.3 for more details.…”

“…For more details on this topic we refer to [18,44] and their references. We mention that elliptic problems associated with the operator L s Ω,p (κ, •) subject to the Dirichlet boundary condition have been investigated in [16,17,25,29] where the authors have obtained some fundamental existence and regularity results. The case of Neumann and Robin type boundary conditions (with κ = 1) is contained in [46].…”

“…The case of Neumann and Robin type boundary conditions (with κ = 1) is contained in [46]. We refer to [25,45] for further results on associated parabolic problems.…”