On the blow-up of solutions to anisotropic parabolic equations with variable nonlinearity |
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Authors: | S Antontsev S Shmarev |
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Institution: | (1) Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK |
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Abstract: | The aim of this paper is to establish sufficient conditions of the finite time blow-up in solutions of the homogeneous Dirichlet
problem for the anisotropic parabolic equations with variable nonlinearity $
u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} }
$
u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} }
. Two different cases are studied. In the first case a
i
≡ a
i
(x), p
i
≡ 2, σ
i
≡ σ
i
(x, t), and b
i
(x, t) ≥ 0. We show that in this case every solution corresponding to a “large” initial function blows up in finite time if there
exists at least one j for which min σ
j
(x, t) > 2 and either b
j
> 0, or b
j
(x, t) ≥ 0 and Σπ
b
j
−ρ(t)(x, t) dx < ∞ with some σ(t) > 0 depending on σ
j
. In the case of the quasilinear equation with the exponents p
i
and σ
i
depending only on x, we show that the solutions may blow up if min σ
i
≥ max p
i
, b
i
≥ 0, and there exists at least one j for which min σ
j
> max p
j
and b
j
> 0. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine
the absorption (b
i
≤ 0) and reaction terms. |
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Keywords: | |
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