Large solutions of parabolic logistic equation with spatial and temporal degeneracies

There is studied asymptotic behavior as t → T of arbitrary solution of equation P0(u) := ut - Δu = a(t; x)u -b(t; x)|u|p-1u in [0; T) × Ω where is smooth bounded domain in ℝN, 0 < T < 1,∞ p > 1, a(·) is continuous, b(·) is continuous nonnegative function, satisfying condition: b(t; x) ≥ a1(t)g1(d(x)), d(x) := dist(x; ∂Ω). Here g1(s) is arbitrary nondecreasing positive for all s > 0 function and a1(t) satisfies: a1(t) ≥ c0 exp(-ω (T - t)(T - t)-1) ∀t〈 T; c0 = const〉 0 with some continuous nondecreasing function ω(T) ≥ 0 ∀T > 0. Under additional condition: →(·) → ω0 = const > 0 as T → 0 it is proved that there exist constant κ : 0 < κ < ∞, such that all solutions of mentioned equation (particularly, solutions, satisfying initial-boundary condition u|τ = ∞, where τ = (0; T) × ∂ω∪[ {0} ×Ω ) stay uniformly bounded in Ω0 := {x ϵ Ω : d(x) > κ∂0 2 } as t → T. Method of investigation is based on local energy estimates and is applicable for wide class of equations. So in the paper there are obtained similar suficient conditions of localization of singularity set of solutions near to the boundary of domain for equation with main part P0(u) = (|u|λ-1u)t - ∑ i N=1(|∇xu|q-1uxi )xi if 0 < λ ≤ q < p.

Авторы
Издательство
American Institute of Mathematical Sciences
Номер выпуска
4
Язык
Английский
Страницы
895-907
Статус
Опубликовано
Том
10
Год
2017
Организации
  • 1 Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Dobrovol'skogo str. 1, Slavyansk, Donetsk region, 84116, Ukraine
  • 2 Peoples' Friendship University of Russia, Miklukho-Maklaya str. 6, Moscow, 117198, Russian Federation
Ключевые слова
Large solutions; Parabolic logistic equation; Spatial degeneracies; Temporal degeneracies
Дата создания
19.10.2018
Дата изменения
19.10.2018
Постоянная ссылка
https://repository.rudn.ru/ru/records/article/record/5398/
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