Behavior of blow-up solutions for quasilinear parabolic equations
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Behavior of blow-up solutions for quasilinear parabolic equations Yevgeniia A. Yevgenieva (Presented by I. I. Skrypnik) Abstract. We study the quasilinear parabolic equation (|u|q−1 u)t − ∆p u = 0 in a multidimensional domain (0, T ) × Ω under the condition u(t, x) = f (t, x) on (0, T ) × ∂Ω, where the boundary function f blows-up at a finite time T , i.e., f (t, x) → ∞ as t → T . For p > q > 0 and the boundary function f with power-like behavior, the upper bounds of weak solutions of the problem are obtained. The behavior of solutions at the transition from the case where p > q to p = q is investigated. A general approach within the method of energy estimates to such problems is described. Keywords. Quasilinear parabolic equation, method of energy estimates, weak solution, singular boundary data.
1.
Introduction We will consider the initial-boundary-value problem: (|u|q−1 u)t − ∆p u = 0,
(t, x) ∈ (0, T ) × Ω,
p, q > 0,
(1.1)
u(0, x) = u0 в Ω, u0 ∈ Lq+1 (Ω), (1.2) u(t, x) = f (t, x) → ∞ as t → T, (1.3) ∂Ω ) ∑ ( where ∆p u = ni=1 |∇x u|p−1 uxi x , T > 0 is called the blow-up time, and Ω is a bounded connected i domain in Rn (n > 1) with C 2 –smooth boundary ∂Ω. The function f determines the singular boundary data. We consider that it admits a continuation in the domain (0, T ) × Ω and satisfies the conditions: 1,p+1 (Ω)); f (t, ·) ∈ Cloc ([0, T ); Lq+1 (Ω)) ∩ Lp+1 loc ([0, T ); W
ft (t, ·) ∈
L1loc ([0, T ); Lq+1 (Ω))
p+1 p−q+1
∩ Lloc
([0, T ); L
p+1 p−q+1
(1.4)
(Ω)).
The main specific feature of problem (1.1)–(1.3) is a blow-up of the boundary function (f (t, x) → ∞ as t → T ). We describe the character of this blow-up as t → T with the function ∫ t∫ ∫ q+1 |∇x f (τ, x)|p+1 dxdτ |f (τ, x)| dx + F (t) := sup 0 0. In [13, Section 6.5], it was shown that, for p < q, the localization of a solution occurs always, irrespective of the character of a blow-up of the boundary function f . Moreover, the blow-up set Ωs remains always on the boundary ∂Ω. In works [7, 21] and in [13, Section 6.3], the case p > q which includes the porous medium equation was considered. It was proved that the localization of a solution u happens under the condition imposed on the character of a blow-up of the boundary function F : F (t) = ω(T − t)
q+1 − p−q
∀ t 6 T,
ω > 0.
(1.8)
In this case, if ω = ω(t) → 0 as t → T , then the domain Ωs = ∂Ω. But if ω = ω(t) → ∞ as t → T , the solution is not localized, and Ωs = Ω. The exactness of the obtained result is confirmed, in particular, by the case of porous medium equation (p = 1, q < 1). For example, the localized boundary function for porous medium equation takes the form [18, Chapter 3]: 1 − 1−q
f (t) = C(T − t)
∀ t 6 T,
C > 0.
(1.9)
It is easy to verify that this corresponds to the character of a blow-up (1.8). In [5, 6] and [13, Section 6.4], the case p = q was considered. It is most hard to be studied, because the equation has no self-similarity in the general case. However, with the help of the method of energy estimates, the conditions of localization were obtained. It was s
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