Abstract
This paper is devoted to studying a nonlocal parabolic equation with logarithmic nonlinearity ulog |u|-fint_{Omega } ulog |u|,dx in a bounded domain, subject to homogeneous Neumann boundary value condition. By using the logarithmic Sobolev inequality and energy estimate methods, we get the results under appropriate conditions on blow-up and non-extinction of the solutions, which extend some recent results.
Highlights
In this paper, we consider the Neumann problem to the following parabolic equation:⎧ ⎪⎪⎨ut – ffl u = u log |u| – u log |u| dx, x∈, t > 0,⎪⎪⎩u∂∂un(x=, 00),= u0(x), x ∈ ∂, t > 0, x∈, (1.1) where⊂ Rn is a bounded domain with smooth boundary, ffl u0 dx = 1 || ́u0 dx = 0 with u0 ≡ 0
⊂ Rn is a bounded domain with smooth boundary, ffl u0 dx u0 dx = 0 with u0 ≡ 0
It is a remarkable fact that a nonlocal parabolic equation with logarithmic nonlinearity does not admit the usual maximum principle and the comparison principle
Summary
We consider the Neumann problem to the following parabolic equation:. u0 dx = 0 with u0 ≡ 0. It is a remarkable fact that a nonlocal parabolic equation with logarithmic nonlinearity does not admit the usual maximum principle and the comparison principle Because of this main difficulty, some most effective methods, such as the method of upper and lower solutions, are not valid here anymore. Inspired by the ideas in [9,10,11,12,13,14,15,16,17,18], the threshold results for the global existence and blow-up of the weak solutions are given by the potential well method, the classical Galerkin method, and the logarithmic Sobolev inequality.
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