Abstract
In this paper a class of nonlocal diffusion equations associated with a p-Laplace operator, usually referred to as p-Kirchhoff equations, are studied. By applying Galerkin’s approximation and the modified potential well method, we obtain a threshold result for the solutions to exist globally or to blow up in finite time for subcritical and critical initial energy. The decay rate of the L 2 norm is also obtained for global solutions. When the initial energy is supercritical, an abstract criterion is given for the solutions to exist globally or to blow up in finite time, in terms of two variational numbers. These generalize some recent results obtained in [Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(9):3283–3297, 2018].
Highlights
We study the global existence and finite time blow-up of solutions to the following parabolic type p-Kirchhoff initial boundary value problem
As a byproduct we show that for any M > d, there exists a u0 such that J(u0) > M and that the solutions to problem (1.1) with u0 as initial datum blow up in finite time
By using the method of approximation, we can still obtain the global existence of weak solutions
Summary
We study the global existence and finite time blow-up of solutions to the following parabolic type p-Kirchhoff initial boundary value problem Han and Li [10] considered the global existence and finite time blow-up properties of solutions to (1.3) with f (x, t, u) replaced by |u|q−1u when p = 2 (under homogeneous Dirichlet boundary condition).
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