On the blow-up of ion-acoustic waves in a plasma

Korpusov, M.O.

Автор: Korpusov M.O.
Журнал: Sbornik Mathematics
Том: 202
Номер: 1-2
Год издания: 2011
Издательство: London Mathematical Society
Местоположение издательства: United Kingdom
Первая страница: 35
Последняя страница: 60
Аннотация: This article investigates the existence of a solution and its blow-up of the Sobolev-type evolution equation ∂2∂t2(Δu−u)+∂∂t(Δu−g(x,u))+Δu+f(x,u)=0,(∗) in a region Ω⊂R3, describing ion-sound waves in plasma with nonlinear sources and strong nonlinear dissipation, under the conditions u|∂Ω=0,u(x,0)=u0(x),u′(x,0)=u1(x).(∗∗) Based on a nontrivial generalization of the well-known Levine method [H. A. Levine, Arch. Rational Mech. Anal. 51 (1973), 371–386; MR0348216 (50 #714); Trans. Amer. Math. Soc. 192 (1974), 1–21; MR0344697 (49 #9436)] and on a priori estimates the author of the article proves the following. (i) Problem (∗)-(∗∗) does not possess a global-in-time strong generalized solution t↦u(⋅,t)∈C2([0,∞);H(1)0(Ω)), meaning that its solution blows up at some T<∞; (ii) Problem (∗)-(∗∗) does possess a local-in-time strong generalized solution for a finite interval [0,T0(u0,u1)] with T0>0, satisfying the destruction property lim¯¯¯¯¯t→T0 Φ(t)=∞, where Φ(t):=12∫Ω|∇u(x,t)|2dx+12∫Ω|u(x,t)|2dx is the wave energy. The latter is obtained by means of reducing equation (∗) to the equivalent second-order Nemytskiĭ-type evolution equation d2dt2A(u)+ddtA(u)+A(u)=ddt(u−Ng(u))+u+Nf(u),(∗∗∗) where the operator A:=−Δ+1 is invertible as a consequence of the Browder-Minty theorem. These are the main results, formulated as Theorem 3 by the author, which are important for applications in plasma physics.
Добавил в систему: Корпусов Максим Олегович

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On the blow-up of ion-acoustic waves in a plasmaстатья