Abstract
In this short note, we consider a nonlocal quasilinear parabolic equation in a bounded domain with the Neumann-Robin boundary condition. We establish a blow-up result for a certain solution with positive initial energy.
Highlights
We consider the initial boundary value problem for a nonlocal quasilinear parabolic equation ut = pu + |u|q– u m( )|u|q– u dx, x ∈, t >, ( . )with Neumann-Robin boundary and initial conditions|∇u|p– ∂u =, x ∈ ∂, t >,∂n u(x, ) = u (x), x ∈, where ⊂ RN (N ≥ ) is a bounded domain with a smooth boundary, m( ) denotes the Lebesgue measure of the domain, pu = div(|∇u|p– ∇u) with p ≥, q > p, u (x) ∈ L∞( ) ∩ W,p( ), u (x) ≡, and u dx =
Niculescu and Rovenţa [ ] considered a more general initial boundary value problem of nonlocal semilinear parabolic equation given by ut =
F |u| dx, x ∈, t >, with the Neumann-Robin boundary condition ( . ), and established a relation between the finite time blow-up solutions and the negative initial energy, when p ≥ and f belongs to a large class of nonlinearities by virtue of a convexity argument
Summary
1 Introduction We consider the initial boundary value problem for a nonlocal quasilinear parabolic equation ut = Blow-up theory for solutions of the initial boundary value problem of parabolic equations with local or nonlocal term has been rapidly developed, and there have been many delicate results. For the relations between initial energy and blow-up solution, see [ – ].
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