Abstract
This paper deals with a semilinear parabolic equation with variable source under the case that the initial energy is less than the potential well depth. We deduce a sharp threshold for blow-up and global existence of solutions. Furthermore, we conclude that the global solution decays as the time goes to infinity.
Highlights
In this paper, we consider an initial boundary value problem for the semilinear parabolic equation with variable exponent:⎧ ⎪⎪⎨ut = u + |u|p(x)– u, x ∈, t >,⎪⎪⎩uu((xx, t) =, ) = u (x), x ∈ ∂, t >, x∈, ( . )where is a bounded smooth domain of RN (N ≥ ), u ∈ H ( ), and p(x) is a continuous and bounded function satisfying< p– := inf p(x) ≤ p+ := sup p(x) < ∗ = N . x∈N – Eq ( . ) has been used to model a variety of important physical processes, such as electrorheological fluids [ ], thermo-rheological flows or population dynamics [, ]
To deal with the variable source, it is convenient to introduce a Lebesgue space Lp(·)( ), defined as the space of measurable functions u in satisfying |u|p(x) dx < ∞. We mention that this kind of Lebesgue space or general Sobolev space with variable exponent and their applications have got a lot of attention, see the monograph [ ] and some recent work [ – ] for instance
Motivated by the above research, in this paper we have the main purpose to look for a sharp threshold for blow-up and global existence of solutions of ( . ) in general case ( . ) and ( . )
Summary
< p– := inf p(x) ≤ p+ := sup p(x) < ∗ = N . To deal with the variable source, it is convenient to introduce a Lebesgue space Lp(·)( ), defined as the space of measurable functions u in satisfying |u|p(x) dx < ∞. We mention that this kind of Lebesgue space or general Sobolev space with variable exponent and their applications have got a lot of attention, see the monograph [ ] and some recent work [ – ] for instance. With the norm u p(·) := u Lp(·) ( ) = inf λ > :
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